Fatigue Crack Initiation and Growth in Ship Structures

Michael Rye AndersenJanuary 1998

Fatigue Crack Initiationand Growth in ShipStructures

Department ofNaval ArchitectureAnd Offshore Engineering

Technical University of Denmark

Fatigue Crack Initiation andGrowth in Ship Structures

by

Michael Rye Andersen

Department of Naval Architectureand Offshore Engineering

Technical University of Denmark

January 1998

Preface

This thesis is submitted as a partial ful�lment of the requirements for the Danish Ph.D.degree. The work has been performed at the Department of Naval Architecture and O�shoreEngineering, the Technical University of Denmark, during the period of September 1994 toJanuary 1998. Professor Preben Terndrup Pedersen and Associate Professor Peter FriisHansen have supervised the study.

The study was �nancially supported by the Technical University of Denmark and thissupport is gratefully acknowledged.

During my study Professor Petershagen gave me the opportunity to visit the Institutf�ur Schi�bau der Universit�at Hamburg, Germany | and Professor Sumi and AssociateProfessor Kawamura arranged for me to visit the Department of Naval Architecture andOcean Engineering, Yokohama National University, Japan. Two very inspiring and fruitfulperiods - I thank the sta� of the departments for their help and kindness during my visits.

Special thanks to all of my colleagues at the Department, friends and family, and espe-cially to my two supervisors, Preben and Peter, for invaluable help and support.

Michael Rye AndersenLyngby, May 19, 1998

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Executive Summary

Fatigue crack initiation and fatigue crack growth are important damage modes in ship struc-tures. An important example is the large number of longitudinal to web frame connections,which make up an essential part of the production costs of a very large crude oil carrier(VLCC). The slot structures are complicated welded details exposed to dynamic loads dur-ing the service of the vessel and, if not adequately designed, signi�cant fatigue crackingmay occur and cause major costs of repair. This was the case of the early cracking of thesecond-generation VLCCs. The objective of the present work has been to present new fa-tigue analysis tools and to develop a new fatigue-resistant slot design which is suitable forapplication of welding robots.

In conventional slot structures, cracks can often be found at the toe and the heel of thebracket used to transfer load from the pressure on the side shell into the web frame. In theproposed new design, the bracket is removed in order to avoid these cracks and to ease theapplication of welding robots. The result is that the load to be carried by the weld seambetween the web frame and the longitudinal is increased and the cutout becomes a hard-spotarea. Therefore, a �nite element based shape optimisation is employed to reduce the stressesin this region and a new shape optimised cutout is formed.

A probabilistic fatigue damage model is established with the purpose of comparing therelative fatigue strength of the new shape optimised slot and a conventional design. Themodel is based on the \quasi-stationary narrow-band" theory, which allows for inclusion ofthe non-linear e�ect of the intermittent submergence of the side shell close to the load lineof the vessel. On the basis of an assumed sailing route, fatigue lives of the two types ofstructures are estimated with the e�ects of di�erent heading angles and manoeuvring in badweather taken into account. The toe of the bracket is predicted to be the weakest pointof the conventional structure with an estimated fatigue life of 5 years, while 28 years arepredicted for the weakest point of the shape-optimised design, i.e. the fatigue strength of thenew structure is increased by a factor of 6. At the same time, manual �tting of the bracketis avoided, which can ease the application of welding robots.

Cracks may initiate at imperfections caused by welding and thermal cutting of edges.Conventional fatigue damage approaches are normally based on the assumption that if sub-sequent crack propagation occurs, it is in a self-similar manner. However, branched crackpropagation can be found in structures containing residual stresses or in details exposed to

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iv Executive Summary

biaxial loading, where curved crack propagation may have a signi�cant in uence on the crackgrowth rate. A two-dimensional multipurpose crack propagation procedure is established inorder to investigate mixed mode crack propagation in welded structures subjected to high-cycle loads. The method is based on a step-by-step �nite element procedure with continuousremeshing of the domain surrounding the propagating crack. Special emphasis is put on thee�ect of residual stresses and \crack closure". The latter is of major importance to crackspropagating in compressive residual stress �elds. The simulation procedure is validated bycomparison with experimental and numerical data found in the literature covering both thein uence of holes, biaxial loading and residual stresses. A good correlation is obtained.

The crack simulation procedure is applied to the conventional and the shape-optimisedslot structures. Residual stresses are estimated on the basis of relatively simple inherentstrain distributions. Cracks, 1 mm in length, are initiated at stress concentrations in thebend of the cutouts and close to the weld between the longitudinal and the web frame. Fromthe simulations, it is observed that compressive residual stresses close the faces of the crackinitiated close to the weld, which reduces the risk of crack propagation. This is an importantobservation because imperfections from welds can act as crack initiators.

The crack simulation procedure is restricted to constant amplitude loading and a repre-sentative cyclic wave is determined for the simulations by the established probabilistic model.However, the stress intensity ranges obtained for the con�gurations with initial cracks in thebends result in prediction of crack arrest at initial stages, when the loads from the equivalentwave are applied. This makes it di�cult to conclude that one of the designs is superior to theother with respect to stable crack propagation. Nevertheless, crack simulations performedwith threshold values equal to zero indicate that the initial stress intensity range for theoptimised cutout is larger than for the conventional cutout, but it drops to zero after a smalldistance of crack propagation. The stress intensity range for the conventional structure startsat a lower range, but it increases continuously as the crack propagates. The same trend isto be expected at a stress level above the threshold limit. Thus, the optimised slot designseems to be superior to the conventional design.

Synopsis

Initiering og v�kst af udmattelsesrevner er betydningsfulde skadestilstande for skibsstruk-turer. Et vigtigt eksempel er det store antal samlinger mellem webrammer og longitudinaler,som udg�r en ikke uv�sentlig del af produktionsomkostningerne for en supertanker. Disseforbindelser er komplicerede svejste detaljer, der gennem skibets levetid er udsat for dy-namiske belastninger. Udbredt revnedannelse kan derfor forekomme, hvis disse detaljer ikkeer passende dimensioneret. Revnedannelse introducerer betydelige vedligeholdelsesomkost-ninger. Form�alet med dette arbejde har v�ret at pr�sentere nye v�rkt�jer til bestemmelseaf udmattelsesstyrke, samt at udvikle en ny forbindelse til gennemf�ring af longitudinaler iwebrammer. Den udviklede forbindelse har en h�j sikkerhed mod udmattelse, og den tilladeren let applikation af svejserobotter.

I traditionelle forbindelser til gennemf�ringen af longitudinaler i webrammer kan revnerofte �ndes ved t�aen og h�len af det stag, som bruges til at f�re trykket fra skibssidenind i webrammen. I det nye design fjernes staget for at undg�a disse revner, samt for atlette implementeringen af svejserobotter. Derved for�ges den kraft, som skal overf�res afsvejses�mmen mellem longitudinalen og webrammen, og udkappet bliver et \hard-spot"omr�ade. En �nite element baseret formoptimeringsprocedure bruges derfor til at reduceresp�ndingerne omkring udsk�ringen, og et nyt optimeret udkap opn�as herved.

En probabilistisk udmattelsesmodel etableres med det form�al at sammenligne udmat-telsesstyrken af det nye design med styrken af en traditionel samlingstype. Udmattelses-modellen er baseret p�a en \kvasi-station�r smalb�andsteori", hvilket tillader inkludering afden ikke line�re e�ekt fra det uregelm�ssige vandtryk der virker p�a skibssiden i omr�adetomkring vandlinien. E�ekter af forskellige m�devinkler, samt man�vrering i d�arligt vejr erligeledes inkluderet. Levetiderne af de to samlingstyper bestemmes p�a basis af en antagetsejlrute. Med en beregnet levetid p�a 5 �ar, forudsiges t�aen af staget at have den kortestelevetid i det konventionelle design, mens det svageste punkt i det formoptimerede udkapforudsiges en levetid p�a 28 �ar. Det nye design er alts�a seks gange st�rkere med hensyn tiludmattelse, og det tillader let anvendelse af svejserobotter.

Revner kan initieres ved defekter som stammer fra svejsning eller termisk sk�ring afkanter. Revnev�kstmodeller er normalt baseret p�a en antagelse om line�r udbredelse afrevnen. Men krum revneudbredelse kan forekomme i konstruktioner, som indeholder resi-dualsp�ndinger eller i detaljer, som er udsat for bi-aksiel last. Den krumme revneudbredelse

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kan have en stor betydning for revnev�kstshastigheden. En to-dimensional revnev�kst-model opbygges derfor til unders�gelse af revneudbredelse i svejste konstruktioner, som erudsat for mangegangsp�avirkninger. Metoden baseres p�a en \skridt-for-skridt" �nite elementprocedure med nyt net i omr�adet omkring revnen for hvert skridt. Der vil blive lagt v�gt p�ae�ekten af residualsp�ndinger og revnelukning, hvor den sidste del er af speciel betydning,n�ar en revne udbreder sig i et residualsp�ndingsfelt med tryksp�ndinger. Proceduren veri-�ceres mod eksperimentelle og numeriske resultater fundet i litteraturen, hvor ind ydelseaf huller, bi-aksiel last, samt residualsp�ndinger er inkluderet. En god overensstemmelseopn�as.

Revnesimuleringsproceduren bruges i et sammenligningsstudie af det nye design og entraditionel forbindelse. Residualsp�ndinger bestemmes p�a basis af relativt enkelte udtrykfor t�jningsfordelinger hidr�rende fra opbygning af konstruktionerne. Revner med en l�ngdep�a 1 mm initieres p�a steder med h�j sp�ndingskoncentration, hvilket er i rundingen afudkappene og ved svejsningen mellem udkappet og longitudinalen. Simuleringerne viser atkompressive residualsp�ndinger lukker revner, som initieres i udkappet t�t ved svejsningen.Dette er en vigtig observation, da revner kan dannes fra defekter i svejsningen.

Revnesimuleringsproceduren er begr�nset til dynamiske laste med konstant amplitude,og en repr�sentativ cyklisk last �ndes derfor ved anvendelse af den probabilistiske model.Den variation i sp�ndingsintensitet, som opn�as for de to kon�gurationer, med initielle revneri rundingen af udkappene, resulterer i revnestop p�a de initielle stadier. Revnesimuleringerudf�rt med gr�nsev�rdien for revnestop sat lig med nul, indikerer, at det initielle sp�ndings-intensitetsudsving vil v�re st�rst for den optimerede detalje, men efter en kort revneudbre-delse vil udsvinget i sp�ndingsintensitet falde til nul for denne detalje. For det traditionelledesign starter sp�ndingsintensitetsudsvinget p�a et lavere niveau, men det stiger efterh�andensom revnen vokser. Den samme trend kan forventes for et sp�ndingsintensitetsudsving overgr�nsev�rdien for revnestop. Den formoptimerede gennemf�ring synes derfor at have st�rststyrke mod udbredelse af lange revner.

Contents

Preface i

Executive Summary iii

Synopsis (in Danish) v

Contents vii

1 Introduction 1

1.1 Overview and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Fatigue Damage Models 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 De�nition of Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Stress Concentration Factors . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Fatigue Damage Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 The S-N Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Cyclic Strain Approach Applied to High-Cycle Fatigue . . . . . . . . 11

2.3.3 Outline of the Cyclic Strain Approach . . . . . . . . . . . . . . . . . 11

2.3.4 Summary of Fatigue Damage Models . . . . . . . . . . . . . . . . . . 15

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3 Probabilistic Fatigue Damage Analysis of Slot Designs 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Probabilistic Fatigue Damage Model . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Direct Load Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Load Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3 The Local Finite-Element Model . . . . . . . . . . . . . . . . . . . . 21

3.2.4 FE Model of the Ship and the Local Stress Transfer Function . . . . 23

3.2.5 Short- and Long-Term Response . . . . . . . . . . . . . . . . . . . . . 26

3.2.6 Fatigue Damage Calculation . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 The Fatigue Life of the Conventional Slot Structure . . . . . . . . . . . . . . 28

3.4 Static Shape Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.2 Result of the Shape Optimisation . . . . . . . . . . . . . . . . . . . . 32

3.5 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Two-Dimensional Crack Propagation, Theory 37

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Basic Linear Elastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . 37

4.2.1 Modes of Crack-Tip Deformation . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Energy Release Rate, G . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Direction of Crack Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Mixed Mode Crack Propagation under Monotonic Loading . . . . . . 43

4.3.2 Mixed Mode Crack Propagation under Cyclic Loading . . . . . . . . 46

Contents ix

4.4 Crack Propagation Models for Constant Amplitude Loading . . . . . . . . . 48

4.5 Mode I Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5.1 Threshold Stress Intensity Range �Kth . . . . . . . . . . . . . . . . . 50

4.5.2 Mode I Crack Propagation Laws . . . . . . . . . . . . . . . . . . . . . 51

4.5.3 Crack Closure and Overloads . . . . . . . . . . . . . . . . . . . . . . 52

4.6 Mixed Mode Crack Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6.1 Threshold Range for Mixed Crack Propagation . . . . . . . . . . . . 55

4.6.2 E�ects of Biaxial Loading . . . . . . . . . . . . . . . . . . . . . . . . 56

4.7 Limitations of the Linear Elastic Fracture Approach . . . . . . . . . . . . . . 56

4.8 Summary of Outlined Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Two-Dimensional Crack Propagation, Simulation 59

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Outline of the Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3.1 Singular Isoparametric Quarter-Point Elements . . . . . . . . . . . . 61

5.3.2 The Displacement Correlation Technique . . . . . . . . . . . . . . . . 62

5.3.3 Stress Intensity Factors by the J-Integral . . . . . . . . . . . . . . . . 62

5.3.4 Procedures for J Computations . . . . . . . . . . . . . . . . . . . . . 63

5.3.5 Evaluation of Numerical Methods for Determination of SIFs . . . . . 68

5.4 Crack Arrest Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5 Extension of the Crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 Updating of Fatigue Life Estimation . . . . . . . . . . . . . . . . . . . . . . 70

5.7 Remeshing of the Crack Domain by the Advancing-Front Method . . . . . . 71

5.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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5.7.2 Meshing Strategy for the Advancing-Front Method . . . . . . . . . . 72

5.7.3 Initialising the Front . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.7.4 Departure Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.7.5 Make New Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.7.6 Update Front/Front Empty ? . . . . . . . . . . . . . . . . . . . . . . 75

5.7.7 Smoothing of the Generated Mesh . . . . . . . . . . . . . . . . . . . . 76

5.7.8 Improvement of the Method . . . . . . . . . . . . . . . . . . . . . . . 76

5.7.9 Application to Cracked Structures . . . . . . . . . . . . . . . . . . . . 76

5.7.10 Examples of Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 77

5.8 Examples of Crack Path Simulation . . . . . . . . . . . . . . . . . . . . . . . 79

5.8.1 In uence of Holes on the Crack Growth . . . . . . . . . . . . . . . . . 79

5.8.2 Non-Symmetric Specimen Exposed to Lateral Force Bending . . . . . 81

5.8.3 Biaxial Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6 Crack Closure 85

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.2 E�ective Stress Intensity Range . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.2.1 Estimation of the Crack Opening Ratio U . . . . . . . . . . . . . . . 87

6.2.2 E�ective Stress Intensity Range under Mixed Mode Conditions . . . . 91

6.3 Crack Closure by the Finite Element Approach . . . . . . . . . . . . . . . . 92

6.3.1 Gap Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3.2 Numerical Modelling of Geometrical Closure . . . . . . . . . . . . . . 95

6.3.3 Comments on Element Modelling of Crack Closure . . . . . . . . . . 98

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Contents xi

7 E�ect of Residual Stresses on Crack Propagation 99

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2 Residual Stress Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2.1 Welding-Induced Residual Stress Fields . . . . . . . . . . . . . . . . . 101

7.2.2 Simpli�ed Prediction of Residual Stresses by Use of Inherent Strain . 104

7.2.3 Residual Stresses from Flame-Cutting . . . . . . . . . . . . . . . . . . 109

7.2.4 Redistribution of Residual Stresses . . . . . . . . . . . . . . . . . . . 109

7.3 Stress Intensity Factors in Residual Stress Fields . . . . . . . . . . . . . . . . 110

7.3.1 Superposition Method . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.3.2 In uence Function/Weight Function Method . . . . . . . . . . . . . . 111

7.3.3 Stress Intensity Factors from Residual Stress Fields by the FEM . . . 111

7.4 Input of Residual Strain and Stress Distributions . . . . . . . . . . . . . . . 113

7.5 Crack Growth Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.5.1 Crack Growth in Transition from Compressive to Tensile ResidualStress Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.6 Test Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.6.1 Crack Located Symmetrically with Respect to the Weld Line . . . . . 115

7.6.2 Crack Located Eccentrically with Respect to the Weld Line . . . . . 116

7.6.3 Mixed Mode SIFs in Residual Stress Fields . . . . . . . . . . . . . . . 116

7.6.4 Crack Propagation along a Butt Weld . . . . . . . . . . . . . . . . . 118

7.6.5 Mixed Mode Crack Propagation in Residual Stress Fields . . . . . . . 119

7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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8 Crack Simulation Procedure Applied to Cutouts 123

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2 Modelling of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.1 Loads and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 124

8.3 Crack Propagation without Residual Stresses . . . . . . . . . . . . . . . . . . 126

8.4 Including the E�ect of Residual Stresses . . . . . . . . . . . . . . . . . . . . 128

8.4.1 Modelling of Residual Stresses . . . . . . . . . . . . . . . . . . . . . . 128

8.4.2 Result of Crack Path Simulation . . . . . . . . . . . . . . . . . . . . . 133

8.5 Comments and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

9 Conclusion and Recommendations for Further Work 137

9.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.2 Recommendations for Further Work . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography 141

A Singular Triangular Quarter-Point Elements 153

B Numerical Examples 155

B.1 Single- and Double-Edge Cracks, Mode I . . . . . . . . . . . . . . . . . . . . 155

B.2 Single-Edge Crack in Mixed Mode Condition . . . . . . . . . . . . . . . . . . 157

B.3 Comparison with Experimental Results . . . . . . . . . . . . . . . . . . . . . 158

Chapter 1

Introduction

1.1 Overview and Background

A large number of complicated welded plate joints can be found in an ocean-going ship.During the service time of the vessel these details are exposed to time-varying loads causedby the irregular seaway, the propulsion system, and changes in the loading conditions. Hardspots, such as connections between longitudinal sti�eners and web frames where local risesin stress intensity can be found, may therefore be prone to fatigue cracking. Thus, newproduction-friendly and more fatigue-resistant designs are always of interest.

The classic brittle fracture of the early welded Liberty ships duringWorld War II triggereda considerable amount of work to prevent fatigue failures. In the intervening half centurysigni�cant insight has been gained in fatigue and fracture mechanisms found in various typesof structures, including ships. Physical models for the description of many common fatiguecracking mechanisms have been developed and experimental work has been carried out toverify and support these models. However, in spite of the e�orts made in the �eld of thissubject, cracking problems frequently occur in ship structures today.

The causes are manifold, but a major reason is that the construction of an ocean-goingship which completely resists fatigue cracking is, if possible at present, economically unfea-sible. Therefore, fatigue cracking of the structural members is to be expected at some levelduring the lifetime of a ship. The acceptable level is determined by the safety demandsset by national and international societies, the skills of the naval architects, and the cost ofrepair of cracks.

Sometimes the acceptable level of fatigue cracking is exceeded when signi�cant structuralchanges are introduced in the ship designs, e.g. when high-tensile steel on a large scale wasadopted in the building of the so-called second-generation of very large crude oil carriers(VLCCs). The higher structural strength of the high-tensile steel is not re ected in the

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2 Chapter 1. Introduction

same improvement of the fatigue strength of the material. Therefore, the reductions ofthe scantlings based on structural strength resulted in increased frequency and severity offatigue cracking in many new tankers (signi�cant fatigue damages were found in the second-generation VLCCs after few years of trade). A large fraction of the cracks observed inthe tankers was in the connections between the transverse web frames and the longitudinalsti�eners in the vertical position between the laden and the ballast waterlines. The non-linear load from intermittent submergence of the side shell makes the details located in thiszone especially prone to fatigue cracking.

The large number of longitudinal to web frame connections constitutes an essential partof the production costs of a crude oil carrier. Especially, when double-hull structures arebuild, where the number of these connections is increased signi�cantly compared to theconventional design, as it can be seen from Figure 1.1. In order to achieve competitive costs,

Figure 1.1: Sketch of typical arrangements of the midship sections for single-skin and double-hull tankers respectively.

some shipyards have made an e�ort to improve the productivity of ship fabrication by theintroduction of robots. Hence, new designs which are more suitable for the use of weldingrobots are of interest. However, before such new designs are applied to locations which areknown to be critical with respect to fatigue, such as longitudinal to web frame connections,it is important to carry out fatigue assessment already at the design stage to avoid cases likethe early cracking of the second-generation VLCCs.

Design tools for the prediction of fatigue lives, of details located at critical points of ships,are therefore important. Models for assessment of the residual strength of details containingan initial crack are also of interest, especially for welded and ame-cut details which oftencontain small imperfections inherent from the manufacture.

1.2. Objectives and Scope of the Work 3

The present study deals with models for both types of fatigue assessments, a probabilis-tic approach for fatigue life estimations of uncracked details and a two-dimensional cracksimulation procedure for the prediction of stable growth of initiated cracks. These modelswill be applied to a conventional and a new slot design for the longitudinal to web frameconnection where the new structure is suitable for the use of welding robots.

1.2 Objectives and Scope of the Work

A new design of the slot structure for the connection between the longitudinal sti�eners andthe web frames forms the starting point of the present work. The objective is to presenttwo new design tools for fatigue life assessment and to investigate a production-friendly slotdesign which allows for the application of welding robots and at the same time maintainsor even increases the fatigue strength of the structure. A slot structure found in an existingVLCC forms the basis for the work. The �rst part of the study deals with

� structural optimisation of a new design for the longitudinal to web frame connectionby use of a static shape-optimisation procedure,

� establishing a probabilistic fatigue damage model for the estimation of fatigue lives.The model is to include the non-linear e�ects arising from intermittent submergenceof the side shell located close to the mean sea waterline,

� and a comparison of fatigue lives estimated for two designs.

Conventional fatigue damage approaches are normally based on the assumption that thecrack propagates in a self-similar manner. However, branched crack propagation can beobserved in structures containing residual stresses or in details exposed to biaxial loading.Curved crack propagation can lead to both acceleration and retardation of the crack growthand in some cases even result in arrest of the crack propagation. The second part of thestudy focuses on the establishment of a multipurpose numerical procedure for the predictionof mixed mode crack propagation in welded plate structures. The model is to include aqualitative estimation of the e�ects of residual stresses arising from the manufacture of thedetails. The solution is based on a step-by-step �nite element procedure with the basiccomponents given as:

� Model for the prediction of the crack propagation direction for a given stress intensitystate at the crack tip.

� Crack arrest criterion.

� A robust mesh generator for continuous remeshing of the arbitrarily shaped two-dimensional domain, which surrounds the propagating crack.


� A relatively simple procedure for a qualitative estimation of residual stress �elds arisingfrom the welding of thin plates. The method is based on the inherent strain approach.

� Empirical estimation of mixed mode crack closure in the presence of residual stresses.

The procedure is restricted to high-cycle fatigue which implies that the principles of linearelastic fracture mechanics can be employed for the description of the stress state at the cracktip and the crack growth mechanisms. The in uence of residual stresses will be concentratedon e�ects from static residual stress �elds, only a�ected by the propagating crack. E�ectsof shakedown and overloads will not be considered in the present work.

1.3 Structure of the Thesis

The contents of the thesis are presented in nine chapters composed as follows. Chapter 2deals with two di�erent approaches for relating a cyclic stress state to the fatigue strengthof the material. First the common S-N approach is outlined, which is followed by a possibleapplication of the cyclic strain approach to high-cycle fatigue, with the latter discussionbased on the work of Andersen [3].

The fatigue damage model is a signi�cant part of a fatigue damage approach, however,other important elements have to be addressed for reliable fatigue life estimation of a criticaldetail. The components of a probabilistic fatigue damage approach are outlined in Chapter 3.The probabilistic model is �rst applied to fatigue assessment of a conventional slot design.Shape optimisation is applied to a new type of slot structure and the fatigue life of theobtained design is estimated by the probabilistic approach, leading to a comparison betweenthe predicted fatigue lives for the two structures. This part of the thesis can also be foundin Andersen et al. [4]

The second part of the thesis focuses on mixed crack propagation controlled by the stressintensity state at the crack tip. Chapter 4 outlines the theory needed for the development ofa crack simulation procedure, while the aspects of numerical implementation are discussedin Chapter 5 together with examples of application.

Crack closure takes place when the two surfaces of the crack faces come into contact.The phenomenon can be found in most types of fatigue crack propagation but it becomes ofmajor importance when cracks are located in compressive residual stress �elds. Chapter 6is therefore devoted to crack closure.

Chapter 7 deals with the inclusion of the e�ects of residual stresses which arise fromwelding and ame-cutting. First, relatively simple procedures for the estimation of residualstresses caused by welding and ame-cutting of thin plates are outlined. This is followed bya description of the implementation of residual stresses in the crack simulation procedure,

1.3. Structure of the Thesis 5

that is input of residual strain and stress �elds and the determination of stress intensityfactors in the presence of residual stresses.

In Chapter 8 the established crack propagation procedure is applied to the conventionaland the optimised slot designs described in Chapter 3.

Finally, in Chapter 9 conclusions and recommendations for further work are presented.

Chapter 2

Fatigue Damage Models

2.1 Introduction

Dynamic stress variations experienced during the service period of a ship can initiate fatiguecracks in details which are inadequately designed, constructed or maintained. Subsequentcrack propagation may cause failure of primary structural members leading to catastrophicconsequences such as massive oil pollution or loss of the whole ship [46, 72]. Fortunately,these cases are rare and usually related to bad maintenance of the vessel. However, even ifthe fatigue cracking does not lead to complete failure, the cost of inspections and repairs andthe consequences of e.g. minor oil pollution due to leakage can be high [70]. Thus, fatiguecracking should, if possible, be avoided or kept at an acceptable level.

Fatigue initiation is a localised phenomenon which strongly depends on the structuralgeometry and stress concentrations. In welded structures such as ships, cracks are knownto initiate at stress concentrations caused by aws from welding procedures and at cutoutsand plate joints where abrupt geometrical transitions cause a local rise in stress intensity.Usually, the fatigue strength of a critical detail can be improved by changing the design ofthe structure. Therefore, in cyclically loaded structures, the designer should evaluate thefatigue strength of details where high stress concentration can be found. Both simple andrelatively complex approaches are available for such evaluations. They normally consist of

� a load history description,

� the response of the structure to the external load,

� and a model for combining the structural response and the material fatigue strength.

� If the structure is exposed to a stochastic load, a fatigue accumulation hypothesis isalso required.

7

8 Chapter 2. Fatigue Damage Models

All components have to be dealt with in a balanced way for reliable fatigue life estimation.

A probabilistic approach for the estimation of the fatigue life will be presented in Chapter3 while this chapter focuses on fatigue damage models which relate the cyclic stress state atthe critical point to the fatigue strength of the material. Before a discussion of these models,a de�nition of commonly used stress measures is given.

2.2 De�nition of Stresses

Most approaches employed for fatigue life assessment are strongly related to the stress stateat the crack tip. A de�nition of three stress measures often applied for fatigue analysis aretherefore introduced. They are: 1) Nominal stress, 2) hot-spot stress, and 3) notch stress.These stress measures are applied in fatigue analysis where the type of stress used for theestimations depends on the problem to be solved and the desired level of accuracy.

Nominal Stress

The nominal stress �nom is de�ned as the stress which can be obtained straightforwardlyfrom the sectional forces the moments and the dimensions of the component by simpleformulas such as Naviers and Grassho�s equations. Increase in stresses which arise fromdiscontinuities in the structural geometry or local notches like welds are to be excluded inthe determination of the nominal stress. A drawback of this approach is that the de�nitionof the nominal stress can be rather di�cult for complicated structural details. The hot-spotapproach as well as the notch stress approach can be applied to overcome this di�culty.

Hot-Spot Stresses

The basic idea in the hot-spot approach is to de�ne the local stress so that a notch e�ectfrom geometrical discontinuities is considered while the increase in stress arising from theactual geometry of a weld is disregarded. This method has been widely applied to fatigueprediction of critical joints in o�shore structures where the modelling of welds can be rathertroublesome. The hot-spot stress �g is calculated as the nominal stress times a hot-spot(structural) stress concentration factor Kt.

Notch Stress

The notch stress �k is the stress which can be found locally at a notch, e.g. at the edge ofa cutout or at the toe of a weld. The di�erence between the hot-spot stress and the notch

2.3. Fatigue Damage Models 9

Figure 2.1: De�nition of stress measures.

stress is that the actual geometry of the notch (e.g. a weld toe) is taken into account in thelatter approach. The rise in stress from a weld can be included by a stress concentrationfactor Kw. The three types of stress measures are illustrated in Figure 2.1.

2.2.1 Stress Concentration Factors

Stress concentration factors (SCFs) for the di�erent notch e�ects may be calculated fromthe theory of elasticity by various methods. Analytical calculations are mostly suited forrelatively simple con�gurations. The �nite element method, on the other hand, o�ers amore versatile method which can be applied to both two- and three-dimensional problems.When the �nite element method is used for determination of the SCFs of details in weldedstructures the choice is between modelling the weld or not. The latter approach results inthe simplest models, but the stress may become singular at plate joints. In these cases,extrapolation procedures have to be employed for determination of the structural SCFs asdescribed in Section 3.2.3. Alternatively, stress concentration factors for typical details inships can be found in fatigue guide books (see e.g. [2, 34]).

All stress risers have to be taken into account in evaluations of the notch stress. Anoverview of notch factors important to ship structures is given in DNV [34] together with alibrary of stress concentration factors.

2.3 Fatigue Damage Models

There are several models for relating the cyclic stress/strain state at the notch or crack tipto the fatigue strength of a material. Two procedures are presented here. First, the oftenused S-N approach and then a possible application of the cyclic strain approach to high-cycle fatigue. The basic assumption in both procedures is that the fatigue performance ofan unnotched small-scale laboratory specimen describes the fatigue endurance of a \real"structure when it is exposed to the same stress history.


2.3.1 The S-N Approach

The S-N approach is based on fatigue tests which describe the relation between the cyclicstress range and the corresponding number of load cycles for failure. The procedure assumesthat it is only necessary to consider the range in the principal stresses when the fatigueendurance is estimated. Thus, in uence of the mean stress level is neglected.

The S-N fatigue test can be presented in a Log-Log diagram called a S-N curve. Thebasic design curve is given as

LogN = Log�a�mLog�� (2.1)

where N is the number of cycles for failure at the stress range �� and m is the negativeslope of the S-N curve. The desired level of safety can be adjusted by the material-relatedparameter Log�a. DNV [34] recommends that the design curve is based on the mean test curveminus two times the standard deviation which results in a 97.6 % probability of survival.

Damage Accumulation

The S-N design curve given by Eq. (2.1) relates the critical number of load cycles N to aspeci�c stress range. Thus, a damage accumulation model is needed if the S-N approach isto be applied to structures subjected to uctuating loads. For practical purposes, the linearPalmgren-Miner rule can be employed. It states that the total damage experienced by thestructure may be expressed by the accumulated damage from each load cycle at di�erentstress levels. The uctuating stresses can be divided into j levels of constant stress and thedamage is calculated as

D =jX

i=1

niNc;i

(2.2)

where ni is the number of stress cycles at load level i and Nc;i is the corresponding criticalnumber of load cycles. Failure is assumed to occur when the damage index D is equal to1.0. The Palmgren-Miner approach assumes that the cumulative damage D has the sameform regardless of the applied stress and that D is independent of the sequence of appliedstresses. This is in general not the case. However, the Palmgren-Miner rule is still one ofthe most accurate models for cumulative damage calculations.


2.3.2 Cyclic Strain Approach Applied to High-Cycle Fatigue

The outlined S-N approach is the most commonly used model for description of the relationbetween the applied load and the fatigue strength, and it is adopted by most classi�cationsocieties. The S-N approach normally yields good results, but 1) it does not take intoaccount local yielding and 2) it neglects the in uence of the mean stress level. The e�ectof the external load-induced mean stress on the fatigue life is not very pronounced if theresidual stresses at the notch are close to the yield stress level, as they often are in weldedconnections. But, for scallops and cutouts with a lower residual stress level, the mean stressmay play an important role. Thus, for estimations of the fatigue life of scallops and cutoutssubjected to high cyclic stresses, it has been proposed to use the cyclic strain approach,which accounts for local yielding and the mean stress level.

The cyclic strain approach has normally been used in connection with low-cycle fatigue.An application of the method to high-cycle fatigue problems was analysed in a case studyby Andersen [3]. Cracks reported in the longitudinal of a 300 m long bulk carrier after 21years of trade were used for the investigation (see Figure 2.2). For this con�guration, fatiguelives were estimated by the cyclic strain approach and subsequently compared with fatiguelives predicted by the S-N approach. Some of the observations and conclusions made in thestudy will be reviewed here starting with an outline of the cyclic strain approach.

Figure 2.2: Location of reported crack.

2.3.3 Outline of the Cyclic Strain Approach

Cyclically loaded elastic-plastic materials often undergo changes which result in a dynamicyield curve di�erent from the static yield curve. The cyclic strain approach is based on thedynamic strain/stress relationship taking into account local yielding at the notch. Thus, thedynamic yield curve has to be determined for the detail of interest either (1) experimentally,(2) by non-linear �nite element analysis, or (3) by approximation formulas, such as theZeeger or the Neuber rule. The latter method is simple and gives good results compared tothe two other methods [41]. Especially the simple Neuber rule can be useful:

�a =�aE

��a;e�a

�2(2.3)


where E is the modulus of elasticity, �a;e is the elastic notch stress amplitude, �a and �a arethe local elastic-plastic stress and strain amplitudes, related by a material law. A commonlyemployed material law for cyclically loaded structures is [42]

�a = g(�a) =�aE

+��aK 0

�1=n0(2.4)

where K 0 and n0 are material constants representing the plastic part of the strain amplitude�a. The local elastic-plastic stress and strain relation can be obtained from Eq. (2.3) andEq. (2.4) as a function of the elastic notch stress amplitude �a;e. Figure 2.3 illustrates thisrelation for a circular hole in a plate of high-tensile steel. If the structure is exposed to very

Figure 2.3: Cyclic yield curve for a hole in an in�nite plate.

high loads and general yielding is present in the total net section, the Neuber rule can bemodi�ed by an additional factor to yield good results [41].

The damage from each load cycle can be found from the well-known W�ohler curves, butwhen the cyclic strain approach is applied, the stress has to be replaced by a special damageparameter. The damage parameter PSWT proposed by Smith, Watson and Topper [119] canbe employed, taking the e�ect of the mean stress into account:

PSWT =q�max �a E =

q(�a + �m) �a E (2.5)

with �max being the maximum stress in each load cycle and �a the local elastic-plastic strainamplitude. Both �max and �a may be obtained by solving Eq. (2.3) and Eq. (2.4). Fornegative mean stress �max should be replaced by �a in Eq. (2.5).

The Palmgren-Miner rule given by Eq. (2.2) is applied in the case of variable amplitudeloading.


Figure 2.4: Sketch of the three di�erent interpretations of the PSWT damage curves.

Observations Made in the Study

The data available for damage curves based on the Smith, Watson and Topper formulationwas reviewed during the study. It was observed that the data in the high-cycle region isrelatively limited for ame-cut edges in steel HF 36. This left some uncertainty how toestablish the damage curves in the high-cycle zone. Heuler [56] has suggested three di�erentinterpretations of the available data:

� \Original", by use of all available data. In the case study, this interpretation wasapplied in the form of a bilinear �t of the data.

� \Elementary", the damage curve is characterised by one exponential distribution, ne-glecting the data in the high-cycle area.

� \Reduced fatigue strength", like the \elementary" interpretation, but with a cuto�level at half of the fatigue strength (the high-cycle point where the PSWT level is equalto half of the \original" PSWT value).

All three interpretations were applied in the investigations together with a deterministicestimation of the load history of the vessel, leading to the very di�erent damage indices D


presented in Figure 2.4. From this �gure, it is seen that the curves based on the two lastassumptions are the most conservative.

It is also important to note that the in uence of the mean stress level was found to bequite signi�cant. Two loading conditions were considered in the case study, a fully loadedand a ballast condition. Only the ballast condition, with a relatively high mean stress level,was found to contribute to the damage index, while the contribution from the fully loadedcondition, with a zero mean stress level, was negligible.

Comparing the S-N and the Cyclic Strain Approach

The S-N approach was applied in the case study using S-N curves recommended by the DNV[34]. The holes were located in a corrosive environment. Nevertheless, the calculations werecarried out for both a corrosive and a non-corrosive environment for comparative reasons,since fatigue data for the cyclic strain approach was only available for non-corrosive condi-tions. Table 2.1 presents the estimated damage indices obtained by the di�erent methods.The results obtained by the S-N approach show good agreement with the two conservativeindices estimated by the cyclic strain approach, despite the fact that the damage estimatedby the cyclic strain approach is from the ballast condition only, while the resulting dam-age index from the S-N approach consists of two equal damage contributions from the twoloading conditions. The damage index obtained by the \original" cyclic strain approachis at least a factor of 10 smaller than the rest of the indices, which indicates that one ofthe conservative methods (or an alternative method) should be used when the cyclic strainapproach is applied for design purposes. The e�ect of the corrosive environment is seen to

Table 2.1: Damage index by di�erent approaches.

Damage approach Damage indexCyclic strain, original 0.11Cyclic strain, reduced fatigue strength 1.66Cyclic strain, elementary 2.13S-N, air 1.17S-N, corrosive environment 2.82

reduce the fatigue life by a factor of 2.5 when the S-N approach is applied. A sensitivitystudy showed that a 10 % higher stress level causes an increase of the damage index by 125% when calculated by the cyclic strain approach, while the same increase would change thedamage index predicted by the S-N approach by approximately 66 %. Hence, the cyclicstrain approach is most sensitive to inaccuracies in the determination of the stress level.


2.3.4 Summary of Fatigue Damage Models

The cyclic strain approach takes into account the e�ect of local yielding and the mean stresslevel, which can be important to the fatigue life predictions of scallops and cutouts. Themethod has been successfully applied to low-cycle fatigue, but the application of the methodto high-cycle fatigue is for some materials encumbered with a level of uncertainty due toa relatively limited number of data points available in the high-cycle region of the PSWT-damage curves. Depending on the interpretation of this data, the procedure has been shownto predict fatigue lives similar to or a factor of 10 less than fatigue lives estimated by theS-N approach [3]. The cyclic strain approach could o�er an interesting alternative to the S-Napproach if more data becomes available in the high-cycle area. However, the S-N approachwill be applied in the following.

Chapter 3

Probabilistic Fatigue Damage

Analysis of Slot Designs

3.1 Introduction

The connections between the longitudinal sti�eners and the transverse web frames have forlong time been regarded as weak points in the fatigue strength of a traditional tanker design,with the fatigue problems becoming especially pronounced in the second-generation VLCCs,which are built with extensive use of high-tensile steel. The connections are complicatedwelded structures with a large number of potential crack initiation sites. In a study ofregistered crack in �rst- and second-generation VLCCs, Yoneya et al. [151] found thatmost of the fatigue cracks are initiated at the toe and the heel of the at bar sti�enersas illustrated in Figure 3.1. An obvious way to avoid these cracks is to remove the at barsti�ener. However, an elimination is not without cost. The load to be transferred by the weldseam between the web frame and the longitudinal is raised, which results in an increased

Figure 3.1: Detail of typical cracks in side structure.

17

18 Chapter 3. Probabilistic Fatigue Damage Analysis of Slot Designs

stress level at the cutout. To overcome this problem, Kamoi [66] introduced a speciallyshaped cutout (apple shape) suitable for stress relaxation in order to keep the stress level inthe slot structure at an acceptable level.

An objective of this study was to design a slot structure which allows for an increasedapplication of automatic welding procedures and, at the same time, reduces the number offatigue cracks. To avoid manual �tting of the at bar sti�ener, a design was sought wherethe sti�ener was removed. But, the elimination of the bracket has some side e�ects, thebuckling strength of the web frame is weakened and the cutout becomes a hard-spot area. Arational way of reducing the stress level is to shape-optimise the cutout. The �nite elementbased optimisation program ODESSY [100] was employed for that purpose and a new shape-optimised slot design was obtained. The buckling strength of the web frame was secured bymounting a vertical sti�ener, which did not introduce new hard-spot areas.

A non-linear probabilistic fatigue damage model was established in order to compare therelative fatigue strength of the shape-optimised slot structure and the original detail. Theprocedure for fatigue life estimation consists of a stochastic generation of the wave climateto be experienced by the vessel during a design life. The ship hull is modelled in the 3-Dline program I-ship (developed by the Technical University of Denmark), and the responseof the ship due to the simulated environmental loads is obtained by a strip program. Whenthe dynamic loads and the ship response are known, it is possible to calculate the global hullgirder loads by use of �nite-element models.A good description of the dynamic stress variations in the vicinity of the slot structureis important to a reliable fatigue life estimation, and a local �nite-element model with arelatively �ne mesh was established for that purpose. A hybrid formulation was employedwhere the local stress response was calculated on the basis of the loads obtained from theglobal analysis.The non-linear damage prediction was based on a quasi-stationary narrow-band model [43]where the short-term response for each simulated sea state was derived by adding the stresscomponents from all global load cases with their correct phase. The S-N approach wasapplied to the fatigue damage calculations with the stress range determined as the di�erencebetween the maximum and the minimum stresses for a load cycle of 2�. The mean damagerate E[�(��)m] was calculated from the weighted short-term peak distributions over allthe sea states and heading directions. Finally, the fatigue life could be estimated on thebasis of the simulated stress range �� and the fatigue accumulation model where the linearPalmgren-Miner rule was applied.

Two main issues are presented in this chapter: 1) An outline of a non-linear probabilisticfatigue damage model and 2) development of a new slot design for the longitudinal webframe connection by application of structural shape-optimisation. These two componentslead to a comparison of estimated fatigue lives for a conventional and a new shape-optimiseddesign. The fatigue life estimation of the conventional design will be used for the outline ofthe probabilistic fatigue damage model.

3.2. Probabilistic Fatigue Damage Model 19

3.2 Probabilistic Fatigue Damage Model

Yoneya et al. [151] registered fatigue cracks in the �rst- and the second-generation VLCCs.Figure 3.2 shows the distribution of observed cracks in the side longitudinals of the second-

Figure 3.2: Distribution of crack found in side longitudinal for second generation VLCCsfrom Yoneya et al. [151].

generation VLCCs with an average of a little over 10 cracks found in each vessel. The �gureshows that in the longitudinal direction, most of the cracks were observed in the midshiptanks while in the vertical direction, the observed cracks were concentrated in the regionwhere the side shell is exposed to non-linear wave pressure going from the mean still waterload line to a level of approximately 8 metres below this line. It should also be noted fromFigure 3.2 that the starboard side has been more prone to fatigue cracking than the portside. This is probably related to the sailing routes and the loading conditions of the vessels.Similar trends of the location of fatigue cracks were found in a study by Schulte-Strathaus[111], who organised registered cracks from nine survey reports into a database. In thisinvestigation, 42 % of all cracks was reported to be located in the side shell. With the

Table 3.1: Main particulars for the double hulled VLCC.

Length overall 343.71 mLength between perpendiculars 327.00 mBreadth moulded 56.40 mDepth moulded 30.40 mDraught laden 19.55 mDraught ballast 10.13 mService speed 14.5 kn


just outlined damage statistics in mind, a slot structure in a midship tank of an existing293,000 dwt double-hulled VLCC was chosen as a basis for the present analysis. The mainparticulars of the vessel are listed in Table 3.1. The detail to be investigated forms part ofthe T-shaped longitudinal number 52 and the web frame number 174, and it is located twometres below the laden load line, as depicted in Figure 3.3.

Figure 3.3: Location of the slot structure.

The fatigue analysis can be split into a number of phases. An overview of the necessarysteps in a fatigue life estimation based on direct probabilistic load generation is given inFigure 3.4. The di�erent components presented in the �gure will be discussed in the followingon the basis of the work of Friis Hansen [43] and the slot structure from the conventionaldesign.

Figure 3.4: Flow diagram for the fatigue analysis procedure.


3.2.1 Direct Load Modelling

The �rst step of direct modelling is to establish a load spectrum representative of the con-sidered lifetime of the vessel. For that purpose the computer program PROSHIP [31] canbe used. Based on the data available for the di�erent Marsden zones (Hs; Tz), PROSHIPsimulates the wave climate for a speci�ed trade. In the present case the route Persian Gulf -Rotterdam was chosen for the analysis. The e�ect of di�erent heading angles and of manoeu-vring in bad weather is handled by PROSHIP. For a detailed description of manoeuvringphilosophy and �t of Marsden zone data, see Friis Hansen [43]. Loading conditions repre-sentative of the entire lifetime of the ship have to be taken into account. For tankers, thefully loaded condition and the ballast condition are normally su�cient. In the present casethe vessel was assumed to operate in each of these two loading conditions for 50 % of thetime. It should be mentioned that the intermediate loading conditions may be expected inthe range from 5 % to 20 % of the lifetime of the ship [43].

3.2.2 Load Transfer Function

The previous section describes the establishment of the environment in which the ship isexpected to operate. The response of the vessel due to the simulated sea states was in thepresent formulation determined by a two-dimensional strip program based on the Gerritsmaand Beukelman formulation. The output of the program is a two-dimensional prediction ofthe ship motion and forces due to a regular sinusoidal wave system with given amplitude aand frequency !. The basic assumptions are:

1. A two-dimensional description of ship motion and forces is su�cient.

2. Slamming e�ects are neglected.

3. The dynamic transients are small and evolve slowly, so that the structure respondsdirectly to the local wave sinusoid without signi�cant e�ect of transients fromprevious waves.

The two-dimensional description of motion and forces might be rather crude, but for thepresent comparative study it is assumed to cover the major part of the subject.

The e�ect of slamming was judged to be relatively low in the midship tank, hence, itis neglected in the fatigue assessment analysis, since the stress ranges at higher probabilitylevels are most important to the fatigue life of the structure.

3.2.3 The Local Finite-Element Model

The local �nite element model contains a longitudinal spacing in the vertical direction of theside shell and a web frame spacing in the longitudinal direction with half spacing on each side


Figure 3.5: Finite element mesh used for the local model of the conventional slot design.

as depicted in Figure 3.5. Shell elements are used for the modelling with the mesh densityconcentrated in the vicinity of the slot edge and at the connection between the bracket andthe longitudinal. In these regions where a high accuracy is desired, the length of the elementsis around 40 % of the plate thickness t as recommended by Mikkola [85].

Figure 3.1 shows that the weld at the sti�ener toe is a potential crack initiator. Thiswas also observed in the �nite element analyses of the slot structure, when the side shellwas exposed to a unit pressure load. The hot-spot approach is used for the determinationof stresses. Hence, the �nite element model does not include the weld geometry and thestresses become singular at the interesting spot. An extrapolation method was thereforeemployed to �nd the approximate stress level at the weld toe. An extrapolation procedureproposed by DNV [34] was selected for the present analysis where the stress concentrationfactor is determined by a linear extrapolation to the sti�ener toe. Two points located att=2 and 3t=2 from the intersection line were used for the linear extrapolation as illustratedin Figure 3.6. The calculated SCF due to the structure Kt was equal to 1.49 and the SCFcaused by the weld Kw was assumed to be 1.5 as recommended by DNV [34].

The slot construction is a part of the hull structure. Thus, it is exposed to a number ofdynamic forces, where the main cyclic load contributions are due to wave-induced

� vertical and horizontal hull girder bending,

� internal pressure exerted by uids in the tanks because of vertical acceleration,

� dynamic sea pressure loads.


Figure 3.6: Extrapolation of the principal stresses to the sti�ener toe.

It is important to the above-mentioned stress contributions to take into account thatthey act with di�erent phases. The e�ect of wave-induced hull girder torsional stresses isneglected, which is usually justi�able for tank ship sections.

A set of boundary conditions for the model was determined by considering the de ectionof a side shell with a forced pressure load. The de ection is asymmetric due to the presenceof the at bar sti�ener. A larger pilot model covering three web frame spacings veri�edthat the boundary conditions shown in Figure 3.7 give a good approximation to the actualde ection, if the web frame and the at bar sti�ener are placed slightly eccentrically in thex-direction.

3.2.4 FEModel of the Ship and the Local Stress Transfer Function

This section concentrates on the global models which form the boundary conditions (loads)for the local detail just discussed. With the assumptions described in Section 3.2.2, the loadcontributions are reduced to:

1. Vertical hull girder bending.

2. Deformation of the web frame due to external sea water pressure.

3. Internal pressure exerted by vertical acceleration of ballast water or cargo oil.

4. Sea pressure on the side shell.

The load cases illustrated in Figure 3.8.

The hull girder bending moments are determined by the strip program and the globalstress distribution is obtained by means of Naviers theory. These stresses are transformedto edge loads for the boundaries with the label B in Figure 3.7.


Figure 3.7: Boundary conditions for local structural model.

A two-dimensional beam model is used for calculating the deformation of the web framedue to external sea water pressure arising from a simulated wave height. Afterwards, thestress distributions are calculated from Grassho�s and Naviers theories. These stresses arealso transformed into edge loads for the boundaries A, C, and D. Similar calculations aremade for the internal pressure exerted by simulated vertical acceleration of the uids in thetanks.

The external sea water pressure is non-linear due to the intermittent submergence ofthe side shell. Because of this non-linearity the fatigue damage of the combined stresscannot be calculated by a traditional frequency domain analysis. However, the stochasticcombination-problem of the non-linear stress from the external pressure and stresses causedby the wave-induced moments can be solved by application of a quasi-stationary narrow-bandmodel [43].

The local sea water pressure shows a high correlation with the relative, instantaneouswave elevation, when the motions of the vessel are accounted for [32]. Given the transferfunctions Zheave and �pitch (from the strip program) the instantaneous wave elevation abovethe mean still waterline Zaw at location xl can be calculated as (by use of a coordinatesystem with origo as depicted in Figure 3.8)

Zaw(xl; a; !) = afZw(xl; !)� fZheave(!) + �pitch(!)xlgg (3.1)


Figure 3.8: Load cases for fatigue analysis (shown only for the fully loaded condition).


where Zw is the incoming wave elevation and small pitch angles are assumed. The localhydrodynamic pressure P (a; !) is calculated as

P (a; !) =

8><>:

0 for zmsw + zaw < zg�fzmsw + zaw(a; !)g for zmsw < z < zmsw + zaw

� gfzmsw + zaw(a; !)ek(zmsw�z)g for z < zmsw ^ z < zmsw + zaw

(3.2)

where g is the gravity, � is the density of the sea water, and zmsw is the mean still waterline.The reduction in pressure because of the depth is modelled as ek(z�zmsw), where j z � zmsw jis the distance from the mean waterline to the point considered. When Eq. (3.2) is usede�ects of di�racted waves on the hull and waves generated by ship oscillations are neglected,which is a reasonable assumption due to wall-sidedness at the section considered [43].

3.2.5 Short- and Long-Term Response

The quasi-stationary model used for prediction of the fatigue damage in the web frame andthe side shell assumes that the wave spectrum is narrow-banded, which is justi�able for shipstructures [43]. For a narrow-band process the stress range is twice the amplitude leadingto the following Rayleigh distributed stress range within each short term condition:

F��(�) = 1� exp

� �2

8 m0

!(3.3)

where m0 is the spectral moment of order zero. When the S-N approach is applied, fatigueanalysis is based on the assumption that it is only necessary to consider the range of cyclicprincipal stress in the prediction of the fatigue endurance. Principal stresses obtained fromeach load case are summed taking into account the correct phase of the stress components.For each short-term sea state, the resulting stress for a given amplitude a and frequency !is determined by:

�(a; !; �) = �hull girder bending(a; !; �) + �local deformation(a; !; �)

+ �internal pressure(a; !; �) + �external pressure(a; !; �) (3.4)

With the narrow-band theory, it is possible to specify the stress range �� for a given waveamplitude a and frequency !. It is determined as the di�erence between maximum andminimum stress within each stress response cycle:

�� = maxf�(a; !; �)g �minf�(a; !; �)g (3.5)


when the phase � is permitted to run through an entire load cycle [0; 2 �].

The mean fatigue damage rate is obtained by unconditioning the short-term peak distri-bution of Eq. (3.5):

E[D] =T

KmatE[�(��)m]

=T

Kmat

ZHs

ZTs

ZA

ZV

Z

Z��(��(a; !; v; �))m

f(a; ! j Hs; Ts)f(v; � j Hs; Ts))f(hs; ts)dad!dhsdtzdvd� (3.6)

where T is the time, � the upcrossing rate, Kmat a material fatigue parameter, f(a; ! j Hs; Ts)is the joint distribution of wave amplitude and frequency within each sea state, f(v; � j Hs; Ts)accounts for the e�ect of manoeuvring in severe sea states. It is not possible to performa closed-form integration of Eq. (3.6). Hence, the integration was evaluated by the MonteCarlo Simulation (MCS), which is a commonly used technique for solving a complex integral.The basic concepts of the method are described in numerous papers and textbooks, e.g. inRubinstein [107].

3.2.6 Fatigue Damage Calculation

The mean damage rate is determined at the critical points by the MSC of Eq. (3.6). Thesimulations are performed with the probabilistic program PROBAN [88] using 10000 sim-ulation points, and the outcome is both the mean fatigue damage E[��m] and the meanupcrossing rate ��. The fatigue life estimation can be based on the linear Palmgren-Minerapproach together with design S-N curves. In the present case the S-N curves recommendedby DNV [34] were used for the analysis. A combination of Eq. (3.6) and Eq. (2.1) leads tothe following expression:

logN = log�a� log(1

��E[��m]) (3.7)

from which the equivalent critical number of load cycles N can be determined, which leadsto a fatigue life prediction of the detail. The material properties log�a and m suggested byDNV [34] are shown in Table 3.2.


Table 3.2: The S-N curve parameters recommended by DNV.Air or with cathodic protection.

S-N Curve Material log�a mIb Welded joint 12.76 3.0IIIb Base material 13.00 3.0

3.3 The Fatigue Life of the Conventional Slot Struc-

ture

The �rst step in the fatigue life estimation of a complicated structure is to identify the weakpoints with respect to fatigue. The di�erent load cases depicted in Figure 3.8 were used todetect the spots with high stress concentrations most likely to be prone to fatigue initiation.Figure 3.9 shows the von Mises stresses at the cutout caused by a unit pressure on the sideshell.

Figure 3.9: Von Mises stress distribution at the cutout of the conventional design.

3.3. The Fatigue Life of the Conventional Slot Structure 29

Figure 3.10: Von Mises stress distribution at the toe of the bracket for the conventionaldesign.

Stress concentrations are found in the bend and at the connection between the longitudi-nal and the cutout (the thin lines represent the longitudinal in the shell element modelling).Both the toe and the heel of the bracket are known to be crack initiators. An analysis withequally spaced elements along the line of attachment between the at bar sti�ener and thelongitudinal showed that the largest stress concentration occurs at the toe of the presentcon�guration. Mesh re�nements were implemented in the region of the bracket toe accord-ing the to recommendations of Mikkola [85], and the von Mises stress pattern depicted inFigure 3.10 was obtained for the load case with a unit pressure acting on the side shell.

The fatigue lives of the spots identi�ed as being most likely to experience fatigue initiationwere for the conventional design estimated by the established probabilistic model and by useof the material data from Table 3.2. The predicted fatigue life for the toe at the connectionbetween the at bar sti�ener and the longitudinal was 5 years, while the estimate was110 years for the weakest point of the cutout which has the label 1 in Figure 3.11. Thecalculations con�rm that the toe of the at bar sti�ener is a weak point with respect tofatigue, which was observed by Yoneya et al. [151].


Figure 3.11: Potential crack locations for the conventional design.

3.4 Static Shape Optimisation

In the new design, the at bar sti�ener was removed to avoid the stress concentrations atthe toe and the heel of the bracket. The load to be carried by the weld seam between thelongitudinal and the web frame was thus increased. Stress concentrations appear aroundthe cutout in the web frame as depicted in Figure 3.12. The main concentrations are foundat the bends, but, unfortunately also at the weld seam connecting the web frame to thelongitudinal. By removing the bracket the maximum stress at the cutout is increased by afactor of 1.6. The �nite element based optimisation program ODESSY [100] was employedto reduce the general stress level in the web frame and thus to improve the fatigue strengthof the construction.

3.4.1 General Considerations

The fatigue damage is caused by loads which act with di�erent phase. But the simulationsfor the conventional slot design showed that the non-linear local sea water pressure accountsfor the majority of expected damage. Thus, the static optimisation of the slot structurecould be based on the local sea water pressure alone (load case 4 in Figure 3.8).

For the new design the stress level at the slot edge is of greatest interest. Due to the freeedge and to the relatively thin web plate, two of the principal stresses are approximatelyzero. In this case the von Mises stress given by

�ref =

s3

2�ij�ij � 1

2�ii�jj (3.8)

3.4. Static Shape Optimisation 31

Figure 3.12: Potential crack locations for the conventional design.

is equivalent to the one principal stress di�erent from zero. Hence, the objective functionfor optimisation is based on the von Mises stress level in the web frame.

The splines describing the shape of the cutout are de�ned by nodal points. During theoptimisation, ODESSY calculates the sensitivity of moving these points in directions de�nedby design variables. The shape of the cutout is then improved by moving the design pointson the basis of the sensitivity analysis. A maximum movement of a nodal point in one stepis prescribed to ensure a stable optimisation, which means that the �nal shape is found byiteration. It should be mentioned that in an ideal shape optimisation every design pointwould be free to move in all directions and the result would be totally independent of theinitial geometry of the cutout. But, experience with the applied optimisation program showsthat some restrictions have to be made to avoid mesh distortion and to reduce the CPU timeto an acceptable level. The �nal shape of the slot therefore to some degree depends on howthe design points and the corresponding variables are de�ned. Di�erent combinations weretested with the de�nition given in Figure 3.13 revealing the best results (the slot structurewith the lowest stress level which is still suitable for application in the production). Thedesign points depicted as dots are allowed to move in the direction given by the designvariables illustrated by the arrows in the �gure.


Figure 3.13: De�nition of design variables for the shape optimisation.

3.4.2 Result of the Shape Optimisation

A simplex procedure was used for the optimisation and the �nal shape of the cutout can befound in Figure 3.14. It is seen that the main stress concentrations are found in the bendof the cutout, away from the welding between the web frame and the longitudinal. This isfavourable because in the bend there is no weld to initiate cracks. Compared to a traditionalrectangularly shaped cutout in a slot design without a at bar sti�ener, the principal stressesfrom a unit pressure load are reduced by a factor of 1.5 at the location of the weld.

Again, �nite element results obtained from the di�erent load cases were analysed to �ndthe potential crack locations in the new design with the detected weak points depicted inFigure 3.15. The lifetime was estimated by the probabilistic model and the predicted fatiguelives of both the new and the conventional design are listed in Table 3.3. It is seen from thetable that the fatigue life of the four hard spots in the new slot structure is of approximately

Table 3.3: Estimated fatigue life.

Conventional Estimated lifetime for point 1 110 yearsslot design Estimated lifetime for point 2 5 years

Estimated lifetime for point 3 34 yearsShape-opti- Estimated lifetime for point 4 28 yearsmised design Estimated lifetime for point 5 30 years

Estimated lifetime for point 6 77 years

the same order of magnitude, which is favourable (special attention has been paid to theweld at points 4 and 5). The estimated lifetime for the hard spot at the toe of the atbar sti�ener is about 5 years for the conventional slot structure, i.e. six times less than thelifetime for the new design.

3.5. Comments 33

Figure 3.14: Optimised slot.

3.5 Comments

First of all, too much emphasis should not be placed on the absolute values of the estimatedfatigue lives. The calculations are believed to cover the major contributions to fatigue of theslot structures and the predictions are therefore well suited for the comparative study of thetwo designs. But, as absolute values, the predictions are questionable due to the stochasticnature of the problem and the assumptions made in the study.

The optimisation of the cutout is based on the assumption that the local sea waterpressure accounts for most of the fatigue damage in the connections between longitudinalsand web frames. Figure 3.16 shows the sum (with the label 1) and the di�erent contributionsto the fatigue damage. On the basis of these results the assumption seems to be veri�ed.It should be noted that the contributions act with di�erent phases and that the damage isnon-linear in the stress range.


Figure 3.15: Potential crack locations for new slot.

Figure 3.16 illustrates also how the expected damage is distributed between the ballastand the laden condition. It can be observed that the laden condition accounts for most(approximately 90 %) of the predicted damage. The analysis was carried out for a slotstructure located in the region where the side shell is exposed to non-linear pressure fromthe outside sea water in the laden condition, di�erent patterns are expected for longitudinalslocated outside this region.

When Kamoi [66] compared his new slot structure with the initial design (slot structurewith a at bar sti�ener) he maintained the same stress level at the cutout. Unfortunately,the optimised cutout (from Figure 3.14) does not show the same excellent result. In thepresent analysis the maximum principal stress was found to be approximately a factor of1.45 times larger than in the initial slot structure. But the overall stress level is still lowercompared to the stress at the toe of the at bar sti�ener.

The points in Figure 3.15 are symmetrically located with respect to the longitudinal.Yet, from Table 3.3 it is seen that the estimated lifetime di�ers by a factor of 2 for thepoints in the bends of the cutout. This is caused by the shear forces in the web framewhich is respectively increasing/decreasing the principal stresses in the bends. Close to thelongitudinal (points 4 and 5) the e�ect of the shear force is minor due to the geometry ofthe cutout.

The buckling strength of the web frame is reduced when the transverse at bar sti�eneris removed. A vertical sti�ener can be mounted to prevent buckling of the web frame (thisis a common way to obtain the required buckling strength). Such a vertical sti�ener is idealfor an automatic welding procedure and it does not introduce new weak points in the fatiguestrength of the slot structure. Equally important, the fatigue life of the slot is not decreased

3.6. Summary 35

Figure 3.16: The sum and the di�erent contributions to fatigue damage.

by the in uence of the vertical sti�ener (a minor improvement can be obtained, e.g. for aspeci�c position of the sti�ener, the estimated lifetime for point 4 was increased from 28 to30 years).

In the present analysis the at bar sti�ener is without a soft toe, but results which arepresented in [151] show little in uence of a soft toe (the SCF due to pressure is only reducedfrom 1.7 to 1.6).

3.6 Summary

The heel and the toe of the brackets found in a conventional longitudinal to web framearrangement are known to be crack initiators. A slot structure found in an existing VLCCformed the basis for a comparative study of a conventional design and a new design wherethe brackets are eliminated. The exclusion of the sti�eners increases the load to be carriedby the weld seams between the longitudinal and the web frame and the cutout becomes ahard-spot area. A �nite element based optimisation procedure was applied to reduce thestress level at the cutout and thus increase the fatigue strength of the structure.


Before the shape optimisation, the main stress concentrations were at the welds whichare known to be crack initiators. The shape optimisation reduced the stress level by a factorof 1.6 in this region and in the new design the main stress concentrations were found in thebends away from the welds.

A probabilistic fatigue damage model was established in order to evaluate the relativefatigue strength of the two structures. The model was founded on the narrow-band theory[43], allowing for the inclusion of the non-linear e�ect of the intermittent submergence of theside shell found in the region close to the load line.

The probabilistic model was employed in the analysis of the two designs. In the conven-tional design, the toe of the bracket was predicted to be the weakest point with respect tofatigue with an estimated lifetime of 5 years, while the predicted lifetimes of hard spots inthe shape-optimised cutout were approximately 30 years, i.e. the fatigue strength of the newstructure is increased by a factor of 6.

From the fatigue life estimations of the slot structures located approximately 2 metresbelow the laden load line, it was observed that the laden condition accounts for 90 % of thedamage with most of the damage being caused by the external sea water pressure acting onthe side shell. Further analysis of the structures can be based on these observations.

Finally, it is quite a drastic structural change to remove the at bar sti�ener, but theoptimised cutout could at a relatively low cost be implemented in the conventional slotdesign. Thus, the probability of cracks at the edge of the cutout would be reduced.

Chapter 4

Two-Dimensional Crack Propagation,

Theory

4.1 Introduction

It is good design practice to estimate fatigue lives of critical details in ship structures atthe design stage. But the models used for the fatigue life predictions are usually based onone-dimensional crack initiation and propagation laws, and they do not cover the occurrenceof crack paths which di�er signi�cantly from co-planar paths. Such branched crack growthcan be observed in structures containing a high level of residual stresses or in details whichare biaxially loaded. In some cases the crack will propagate into zones of very high or verylow stress intensity and the crack growth will either be accelerated or it will be retarded and,if the stress intensity becomes su�ciently low, crack arrest might even occur. Curved crackgrowth may thus a�ect the fatigue life of a structure. Hence, the author decided to establisha two-dimensional crack propagation procedure in order to investigate how mixed modecrack growth may a�ect the fatigue strength of the conventional and the shape-optimisedcutouts. This chapter outlines the theory needed for the development of a mixed mode crackpropagation procedure in the frames of linear elastic fracture mechanics, while the aspectsof numerical implementation are discussed in the following Chapter 5.

4.2 Basic Linear Elastic Fracture Mechanics

Fracture mechanics seeks to establish the local stress and strain �elds around a crack tip interms of global parameters such as the loading and the geometry of the structure. Usually,fracture mechanical approaches are divided into 1) linear elastic solutions and 2) non-linearmethods. Only the �rst approach is discussed here. The present formulation is also restrictedto long cracks subjected to constant amplitude cyclic loading (a distinction between long

37

38 Chapter 4. Two-Dimensional Crack Propagation, Theory

Figure 4.1: The three modes of cracking.

and short cracks for normal engineering structures is sometimes assumed to be about 1 mmin length according to Tanaka [130], but it depends on the dimensions of the structure.)

4.2.1 Modes of Crack-Tip Deformation

For linear elastic materials, Irwin [62] suggested describing the stresses in the vicinity of thecrack by stress intensity factors (SIFs). There are basically three di�erent types of SIFs,each describing one of the deformation modes illustrated in Figure 4.1. The superpositionof these three modes forms the general case of cracking. The deformation modes can becharacterised as follows:

� Mode I, is the in-plane tensile mode where the crack surface is symmetrically opened.

� Mode II, is the sliding or shear mode which is present when the crack is exposed toskew-symmetric in-plane loading.

� Mode III, is an anti-plane mode where the crack surface is twisted by forces perpen-dicular to the crack plane.

In simulations of crack propagation in a plate structure, it is usually the combination ofmode I and mode II which is of greatest interest. Irwin analysed this in-plane mixed modeproblem, and by use of Westergaard stress functions [147] he found an analytical solution

4.2. Basic Linear Elastic Fracture Mechanics 39

Figure 4.2: The polar coordinate system.

for the stress distribution in the vicinity of the crack tip:

�11 =KIp2�r

cos�

2

1� sin

�

2sin

3�

2

!� KIIp

2�rsin

�

2

2 + cos

�

2cos

3�

2

!+O(r1=2)

�22 =KIp2�r

cos�

2

1 + sin

�

2sin

3�

2

!+

KIIp2�r

sin�

2cos

�

2cos

3�

2+O(r1=2)

�12 =KIp2�r

sin�

2cos

�

2cos

3�

2+

KIIp2�r

cos�

2

1� sin

�

2sin

3�

2

!+O(r1=2)

(4.1)

where the stresses are given in the polar coordinate system illustrated in Figure 4.2 andhigher order terms are neglected. Eq. (4.1) is therefore only valid in the region close to thecrack tip (r � a). It is seen that the magnitude of the crack tip stresses is controlled bythe stress intensity factors KI; KII, while the distribution of the stresses is governed by theposition relative to the crack tip given by r and �. It is also observed that the distributionof the stresses is singular for r = 0, which is only true for fully brittle materials. That isthe main limitation for a linear elastic approach, since most materials exhibit some state ofplasticity at the crack tip.

Apart from the stresses, the displacements are also controlled by the stress intensityfactors [54]:

u =1

4�

rr

2�

"KI

(2�� 1) cos

�

2� cos

3�

2

!�KII

(2�+ 3) sin

�

2+ sin

3�

2

!#

v =1

4�

rr

2�

"KI

(2�+ 1) sin

�

2� sin

3�

2

!�KII

(2�� 3) cos

�

2+ cos

3�

2

!# (4.2)


here � is the shear modulus of the material, and � is given by

� =

(3� 4� for plain strain,(3� �)=(1 + �) for plain stress

(4.3)

4.2.2 Stress Intensity Factors

Eq. (4.1) shows that the stress intensity factors are the fundamental quantities describingthe singular elastic stresses. The stress intensity factors are functions of the length andorientation of the crack, the geometry of the body, and the applied load distribution. Forthe uniaxially loaded case, the mode I stress intensity factors can be calculated from

K = �p�aF (4.4)

where F takes into account the e�ects of geometry, the crack length and the applied load.F is equal to 1.0 for a crack located in a uniformly loaded in�nite plate perpendicularto the loading direction. Analytical and empirical expressions for stress intensity factorsare collected in compendiums such as Tada et al. [128] and Sih [115] for specimens ofsimple geometries with various crack con�gurations and load combinations. Unfortunately,structural members used in practical engineering are not generally of simple geometry andthey are exposed to complex loading conditions, so speci�c analysis must be resorted to.In such cases, di�erent methods are available, but for complex mixed mode conditions, the�nite element methods seem to o�er the best solutions due to their exibility and ease ofcomputerisation. Methods for determination of SIFs by use of the �nite element are describedin Chapter 5.

4.2.3 Energy Release Rate, G

The basic concept of the energy release method was established in 1921 by Gri�th. For anideally brittle material, he stated that crack propagation will occur if the energy supplied atthe crack tip is equal to or greater than the energy required for crack growth. This criterionwas later reformulated by Irwin [62] by the introduction of the energy release rate, which isa measure of the energy available for an increment of crack extension:

G = �d�dA

(4.5)

where � is the potential energy and A is the released crack area. Crack initiation occurswhen the energy release rate reaches a critical value called the fracture toughness of the

4.2. Basic Linear Elastic Fracture Mechanics 41

material Gc. The fracture toughness is considered a material parameter, and theoreticallyit should be possible to determine this parameter by calculating the energy needed for thenucleation of the atomic bonds in the material. However, materials contain imperfectionsand crack propagation therefore starts before the theoretical value of the fracture toughnessis reached (for steel the experimental value of Gc is about half of the theoretical Gc-value).Thus, instead of using the theoretical approach the fracture toughness of a material is usuallydetermined by standardised experiments, such as the compact tension (CT) specimen test(described in [23, 105]).

A very simple relation is often used for linear elastic materials to describe the relationbetween the fracture toughness and the stress intensity factors in the mixed mode condition:

G =K2

I

E 0+K2

II

E 0(4.6)

where E 0 is the modulus elasticity E for the plane stress and E=(1� �2) for the plane straincondition. Eq. (4.6) describes a circle in the stress intensity factor plane, but investigationshave shown that the critical G-curve is better described by an ellipse [23]. This indicatesthat Eq. (4.6) should be modi�ed by a material dependent constant k to obtain a better �tto the experimentally observed data:

G =K2

I

E 0+

K2II

E 0k2(4.7)

The critical mode I stress intensity factorKIC is sometimes used as a measure for the fracturetoughness of the material, and it can also be determined by experiments. The relationbetween Gc and KIC is obtained through Eq. (4.6) with KII equal to zero. From above itcan be seen that the fracture toughness of the material is a function of the stress intensityfactors but, unfortunately, it is also a function of the stress state (Eq. (4.6) is a functionof E 0). This means that G has to be related to the present stress state, and experimentstherefore have to be carried out for both plain stress and plain strain conditions. A wayof avoiding that for through thickness cracks is to make sure that plain strain is present inthe major part of the crack surface. By analysing the size of the yield zone at the crack tip(ryield � (KIC=�yield)

2=(6�) for plain strain), the following expression has been established:

B > qK2

IC

�2yield(4.8)

where B is the thickness of the specimen and �yield is the yield stress of the material. Theparameter q di�ers from material to material, but an ASTM standard suggest, that q equalto 2.5 is to be used, which results in a plastic zone with a radius � B=50. The minimumcrack length has to be su�ciently large relative to the plastic zone at the crack tip to ensure\brittle material behaviour". This is generally the case when the crack length meet the samedemand as made on B in Eq. (4.8).


4.3 Direction of Crack Propagation

The crack propagates given that enough energy is supplied at the crack tip. It propagatesin di�erent ways depending on the stress state in the vicinity of the crack tip. The simplestcase is mode I fatigue where the crack extends straight along a preferred direction due tosymmetry. This type of fatigue has been the subject of investigations for several decadesand is now quite well established in the frames of linear elastic fracture mechanics. However,structures often contain randomly located cracks which are in a mixed mode stress �eld byvirtue of their orientation with respect to the loading direction. A characteristic of mixedmode (and mode II) fatigue cracks is that they tend to propagate in a non-self-similar mannerresulting in changes in the crack propagation direction during the loading period.

Several attempts have been made to establish a criterion for mixed mode fatigue growth.Recent critical reviews of proposed methods are given in [21, 51, 96]. No attempt will bemade to repeat their work, here only a few of the commonest approaches are reviewed anddiscussed. Most of the currently used criteria for prediction of mixed mode fatigue crackpropagation are initially developed for monotonic loading and the review starts thereforewith the criteria for monotonic loading. The typical con�guration for this examination is aplate with a crack which is inclined at the angle � with respect to the far �eld load �0. Thenomenclature is given in Figure 4.3.

Figure 4.3: Typical con�guration of the angled crack problem.

4.3. Direction of Crack Propagation 43

4.3.1 Mixed Mode Crack Propagation under Monotonic Loading

Maximum Tangential/Opening Stress Criterion (Max �� or MTS)

One of the �rst attempts to predict the crack propagation direction for a plane stationarymixed mode condition was developed by Erdogan and Sih [40]. They investigated the fractureextension in a large plate of brittle material containing an inclined crack. The plate wasexposed to a general in plane loading condition. It was proposed that the crack wouldpropagate from the crack tip in a radial direction de�ned by the plane perpendicular to thedirection of greatest tension. Thus, the crack would extend in the direction of the largesttangential stress ��. In polar coordinates the tangential stress is given by

�� =1p2�r

cos�

2

"KI cos

2 �

2� 3

2KII sin �

#(4.9)

The direction of the maximum tangential stress can be found by setting the derivative ofEq. (4.9) equal to zero. The non-trivial solution is given by

KI sin(�) +KII(3 cos(�)� 1) = 0 (4.10)

This equation can be solved

�0 = � cos�1

8<:3K

2II +KI

qK2

I + 8K2II

K2I + 9K2

II

9=; (4.11)

In order to ensure that the opening stress associated with the direction of the crack extensionis maximum, the sign of �0 should be opposite to the sign of KII [113] (the two possibilitiesare illustrated in Figure 4.4).

Figure 4.4: Sign of the propagation angle.


The criterion for the crack extension taking place was formulated as

Tc =p2r� ��j�=�0 = constant (4.12)

where Erdogan and Sih [40] suggested that

Tc = KIC (4.13)

From experimental observations Erdogan and Sih concluded that Eq. (4.13) should only beregarded as a practical design tool giving conservative results.

The shortcomings of the maximum tangential stress criterion are: a) the normal stressat the crack tip is singular in all directions when linear elastic fracture mechanics is appliedwhich makes the criterion non-physical, and b) the hypothesis is based on one stress �eldcomponent while the rest of the components are ignored [5]. The MTS criterion has beenwidely applied because of its simplicity and supported by many experiments, but work whichdoes not support this criterion can also be found, see e.g. [106, 129].

Strain Energy Density Theory (Min S or SED)

The strain energy density theory proposed by Sih [116] is an often used method to predictthe crack extension angle. The idea is to examine a small element outside a core regionde�ned by r0 (see Figure 4.5) and then base estimation of the crack extension on the stress�eld outside the core region. It is assumed that the energy per unit area (dw/dA) of theelement can be expressed through

dw

dA=S

r(4.14)

where the radius r (see Figure 4.5) is small in comparison with the crack length, and S isthe strain energy density factor representing the intensity of the 1=r energy �eld [116]. Byanalysis of a crack extension through the core area the criterion was formulated:

1. Crack initiation occurs in the radial direction �0, along which the local energydensity possesses a relative minimum Smin:

@S

@�= 0;

@2S

@�2> 0

!for � = �0 (4.15)

2. Unstable crack growth occurs when Smin reaches a critical value Sc, that is:

Smin = Sc (4.16)


Figure 4.5: The core region.

For an isotropic and homogeneous material subjected to a plane mixed mode stress �eld,the strain density can be found from [117]:

S = a11K2I + 2a12KIKII + a22K

2II (4.17)

in which the coe�cients aij are given by

a11 =1

16�[(1 + cos �)(�� cos �)]

a12 =1

16�sin �[2 cos � � (�� 1)]

a22 =1

16�[(�+ 1)(1� cos �) + (1 + cos �)(3 cos � � 1)]

(4.18)

where � is the shear modulus and � is given in Eq. (4.3). Chang [25] investigated the strainenergy density theory (S-theory) for an elliptic and a slit crack con�guration. He found thatthe S-theory is very sensitive to the r0=a ratio (r0=a is the distance from the crack tip to thepoint where the strain density is calculated divided by half the crack length) when appliedto an elliptic crack model, but that it provides predictions which are comparable to othermodels when it is applied to a slit crack con�guration.

Zhengtao and Duo [153] proposed an alternative fracture criterion assuming that thecrack would propagate in the direction of the nearest distance from the crack tip to iso-strain-energy-density curves in the vicinity of the crack tip. This also leads to the criterion describedby Eq. (4.15), but, as the authors state, it is on a more physically based background.

Maximum Energy Release (Max G)

Hussain et al. [58] and Palaniswamy and Knauss [91] independently developed the maximumenergy release criterion by analysing the in uence of a small virtual extension to an existingcrack in the direction �. The criterion for crack propagation was based on maximising theenergy release rate G with respect to the propagation angle �. The criterion reads:


1. Crack initiation occurs in the radial direction �0, which processes the maximumenergy release rate G:

@G

@�= 0;

@2G

@�2< 0

!for � = �0 (4.19)

2. Unstable crack growth occurs when G reaches a critical value Gg, that is:

G = Gg (4.20)

As mentioned earlier, the critical value of G can be found experimentally or by settingKII equal to zero in Eq. (4.7) if KIC is known.

Crack Propagation Angles by the Outlined Methods

Figure 4.6: Comparison of path-directional criteria.

Ramulu and Kobayashi [99] compared crack extension angles predicted by the three presentedmethods. They worked on a uniaxially loaded plate with a crack which was inclined withrespect to the load direction. The KI /KII -relation was used as the free parameter and theresults are shown in Figure 4.6. It can be seen that for a Poissons ratio equal to 0.3 (�steel), the methods give very similar results for small values of the incline angle �0 (the angleis de�ned in Figure 4.3).

4.3.2 Mixed Mode Crack Propagation under Cyclic Loading

At present, no �rmly established cyclic mixed mode criterion exists for the prediction of cycliccrack propagation angles. It is therefore generally accepted that the criteria developed for


monotonic loading are extended also to cover cyclic loading. This is so, despite the fact thatdistinct di�erences in the characteristics of the crack growth trajectory have been observedfor static and cyclic mixed mode crack propagation.

Mageed and Pandey [77] carried out experiments on uniaxially loaded (� = 0.0 in Figure4.3) aluminium alloy sheets with a centre oblique crack con�guration to investigate staticand cyclic mixed mode crack initiation and crack paths. The experimentally obtained crackpaths were compared to crack paths estimated by two of the criteria discussed in the previoussection (and a third criterion called the maximum-normal-strain theory (MTSN), whichwill not be included in the present discussion). Figure 4.7 shows crack paths for bothstatic and cyclic crack growth where the initial cracks are inclined by two di�erent angles

Figure 4.7: Experimental and predicted crack paths for � = 150 and � = 600 (from [77]).

� (the � is de�ned in Figure 4.3). In the mode II dominant case (� = 150), there isa considerable di�erence between the cyclic initiation angle and the angles predicted bythe theoretical criteria, while the static initiation angle is well described by the theoreticalapproaches. However, di�erences between the static experiments and the angle predicted bythe theories start to occur as the crack propagates. When the mode I stress intensity factordominates, crack paths predicted by the various approaches are in fair agreement with thecyclic experiments. Mageed and Pandey [77] concluded that for � < 30O the agreement isnot very good at the initial stage for cyclic loading. Other investigations have also shown thedi�culties in predicting the initial crack angle under dominant mode II loading [96, 106].


It should also be mentioned that mixed mode or mode II crack propagation can causedirectional instability at either very small loads or at large loads close to the yield stressesof the material [99]. In the case of signi�cant yielding, the cracks tend to follow planes ofmaximum shear stress rather than grow in maximum mode I [93, 118]. But, in lieu of betterapproaches and given that the crack usually turns to propagate under mode I conditions[45, 60, 71, 79, 93, 96], the criterion outlined in the previous sections will therefore be usedfor determination of cyclic initiation and propagation angles.

4.4 Crack Propagation Models for Constant Ampli-

tude Loading

Fatigue may be de�ned as a process of cycle-by-cycle accumulation of damage in a materialundergoing uctuating stresses and strains [2]. The main goal of crack propagation modelsis therefore to relate the material damage to the cyclic loads applied. A fatigue processundergoes several stages and from an engineering point of view it is convenient to divide thefatigue life of a structure into three stages [50]:

I Fatigue crack initiation.

II Stable crack propagation.

III Unstable crack propagation.

The �rst of the three stages covers the crack initiation and the early crack growth state,which includes cyclic plastic deformation prior to crack initiation, initiation of one or moremicroscopic cracks and coalescence of these micro-cracks to form an initial macro-crack [51].Notch stress-strain analysis and low-cycle fatigue concepts can be used for the estimation ofthis initiation period. The second period is characterised by stable crack growth, while atthe last stage the crack growth accelerates due to interaction between fatigue and fracturemechanisms. The duration of the third period is usually quite short and it can withoutgreat loss of accuracy be neglected in the fatigue life estimations. In welded and ame cutstructures it is common that aws and defects exist, and the crack initiation period is inthe present formulation taken into account by assuming that the cracks have an initial cracklength a0. Attention will hereafter be paid to the stage of stable crack growth.

4.5 Mode I Crack Growth

The lifetime of structures with initial cracks is very closely related to the crack propagationperiod, since the initiation time is essentially short. Current experimental and theoretical

4.5. Mode I Crack Growth 49

Figure 4.8: Variation of crack propagation rate.

linear elastic approaches try to describe the stable and unstable crack growth by a crackpropagation rate. This rate is de�ned as the incremental crack growth da, divided by theincremental number of load cycles dN , as a function of the stress intensity factor range �Kduring a load cycle. The SIF range is determined as KI,max�KI,min. A schematic illustrationof a crack growth rate curve is given in Figure 4.8 with characteristics of the curve takenfrom [103]. The shape of the curve suggests that the crack propagation period is dividedinto three regions:

� Region I bounded by the threshold value where the crack growth rate goes asymptot-ically to zero as �K approaches �Kth. Below �Kth the crack remains dormant orpropagates at crack growth rates which cannot be determined experimentally.

� Region II is where stable crack propagation takes place which can be described bylinear relationship between log da/dN and log �K.

� Region III crack growth is described by a rapid increase in the crack growth rate goingtowards \in�nity" as the maximum value of the cyclic stress intensity factor Kmax

reaches the fracture toughness of the material KIC.

Ritchie [103] reported that the variation of the crack growth curve (for steels) can becharacterised in terms of di�erent primary fracture mechanisms (see Figure 4.8). In regionII the failure is generally governed by a transgranular ductile mechanism where the in uenceof microstructures and thickness of the specimen stress is minor. Ritchie [103] reported also


the e�ect of the mean stress to be minor, but signi�cant in uence of the mean stress can befound in the case of crack closure (discussed in Chapter 6). In the third region, when Kmax

approaches KIC, the crack growth becomes sensitive to both microstructures and mean stressdue to the occurrence of static fracture modes such as cleavage and inter-granular fracture.Similar strong e�ects of mean stress and microstructures on the crack growth rate can befound in region I, where the in uence of the environment can also be signi�cant. A detaileddiscussion of factors a�ecting the crack growth is given in [103]. Special attention will begiven to the threshold region in the next section.

4.5.1 Threshold Stress Intensity Range �Kth

By integrating the crack growth curve over the critical crack length ac (ac is de�ned as thecrack length causing failure of the structures/components), it is possible to obtain the totalnumber of loading cycles resulting in failure for a given structure/component. The crackgrowth in region I can therefore play an important role, since the amount of time spent inthis region may be predominant for high-cycle fatigue.

The threshold stress intensity factor range �Kth was initially assumed to be a materialconstant, but numerous studies have shown that it depends on several parameters. A recentreview on fatigue thresholds for long-crack fatigue in metallic materials is given by Hadrboletzet al. [53]. They divided the factors a�ecting the threshold stress intensity range intoextrinsic and intrinsic parameters. The extrinsic parameters cover:

� Mean stress/stress ratio e�ect, which again was believed to be in uenced by materialgrain size and the closure of the crack surfaces.

� The e�ect of overloads (not dealt with in the present formulation) where an overloadpeak results in a retardation of crack growth due to several mechanisms, e.g. plastic-induced closure and residual stresses.

� Temperature related to changes in material parameters such as Youngs modulus E andthe yield strength �y. E, �y and Kth usually decrease with increasing temperature,but in special cases an increase in Kth was observed, which was explained by closureof crack faces due to corrosion.

� The environment where formation and dissolution of surface layers, closure of cracksurfaces due to corrosion and incompressible uids in the crack are known to be im-portant e�ects.

The intrinsic parameters include:

� Microstructural features, such as grain size, single- and multi-phase microstructures,porousness, solid solution e�ects, and dislocation arrangements.

4.5. Mode I Crack Growth 51

The conclusion drawn from the review was that the many intrinsic and extrinsic variablesdo not permit a quanti�cation of the threshold value derived from basic principles. For adetailed discussion of the parameters listed above, see Hadrboletz et al. [53]. The numbersof a�ecting parameters may, however, be reduced by the introduction of an e�ective fatiguethreshold stress intensity de�ned by

�Kth;e� = Kth;max �Kcl (4.21)

where Kth;max is the maximum stress intensity and Kcl is the stress intensity at which crackfaces close (investigation of crack closure is the subject of Chapter 6). �Kth;e� may beobtained from experiments and �Kth;e� is reported to be in the range 2.4-2.6 MPa m1=2 forsteel and in the range 0.9 - 1.9 MPa m1=2 for aluminium alloys [53].

Tanaka and Soya [132] analysed the crack growth behaviour of six grades of steel (�y inthe range 163 - 888 MPa) under various stress ratios. They found that all the metals couldbe characterised by the same �Kth;e�, which was outside the range given in [53], namely 3.45MPa m1=2. An expression for �Kth was also suggested which was found to be a function ofR and �Kth;e�:

�Kth = Max[(�Kth;e� +K0)(R0 � R);�Kth;e�] (4.22)

where K0 and R0 are material constants. Their experimental results indicate that Kth iswell described for both R > 0 and R < 0.

4.5.2 Mode I Crack Propagation Laws

The Paris law is a simple, but very often used model for description of the crack growth ratein the linear region II. It reads:

da

dN= C�Km (4.23)

where �K is the stress intensity factor range and c;m are material constants. For mostmaterials m is between 2 and 7, while c is more material-dependent. A shortcoming of theParis law is that it neglects the in uence of the peak stress and the threshold range.

Attempts have been made to complete the description of the crack growth curve. Apopular formula which incorporates the threshold range is:

da

dN= C [�Km ��Km

th] (4.24)


where the factors a�ecting the threshold range (discussed in 4.5.1) are taken into accountwhen �Kth is inserted. This equation is valid for region I and II but does not cover unstablecrack propagation. The Forman equation is often employed to model the crack growth ratein the regions of stable and unstable crack propagation:

da

dN=

C�Km

(1�R)KIC ��K(4.25)

where the in uence of the peak stress is accounted for by the stress ratio coe�cient R(�min=�max). Other modi�cations have been proposed. A list of nine proposed models isgiven in [130]. But, with the outlined formulas, it is possible to describe the whole stressintensity factor range.

4.5.3 Crack Closure and Overloads

The closure of crack surfaces during cyclic loading may have signi�cant in uence on fatiguecrack growth behaviour. Contact between the crack faces may be experienced in manysituations as a consequence of e.g. residual compressive stresses, bending loading, crackasperities or roughness and plasticity [18]. A way of dealing with closure is to apply theconcept of \e�ective" stress intensity range �Ke�, which was introduced in Section 4.5.1.Outside the threshold region the e�ective stress intensity range is determined by replacingKth;max with the maximum stress intensity factor Kmax in Eq. (4.21). The details of \crackclosure" will be discussed in Chapter 6.

This formulation is restricted to constant amplitude loading, and the e�ect of plasticityinduced by overloads (and underloads) and other sequence e�ects are therefore outside thescope of this work. But for completeness it should be mentioned that an overload may causeretardation of the crack growth. Figure 4.9 shows a schematic representation of a crack curveversus the applied load. The retardation illustrated in the sketch can be explained by thefact that an overload may cause the tensile stress at the crack tip to exceed the yield limitof the material which creates an overload plastic region at the crack tip. When the cracktip is relieved, a zone of compressive residual stresses is formed which tends to close thecrack surface and thus causes retardation of the crack growth of the subsequent load cycles.The crack growth is resumed when the crack has propagated out of the compressive residualstress region. Relatively simple analytical approaches like the Wheeler model [148] can beused to estimate the retardation e�ect of overloads. But in some cases these approacheseven fail to model the retardation of a single (mode I) overload. Thus, more sophisticatedapproaches are needed for reliable modelling of the sequence e�ects of variable amplitudeloading. Discussions of variable amplitude loading can be found in e.g. Dominguez [35] andIbs�[59].

4.6. Mixed Mode Crack Growth 53

Figure 4.9: Overload e�ect on fatigue crack growth. The upper curve shows the load historyand the lower shows the crack growth with intermittent overloads.

4.6 Mixed Mode Crack Growth

The growth of cracks under mode I and mode II was �rst systematically studied by Iida andKobayashi [60]. To obtain a model for plane mixed mode crack propagation, they carriedout experiments on a sheet of aluminium alloy exposed to a uniform tensile load. A centralcrack was inclined at an angle to the loading direction. The results of their experimentsshowed that even a small �KII contribution would signi�cantly increase the crack growthrate. However, they also observed that the crack tended to grow in the direction of minimumKII. An observation which has been supported by numerous investigations, see e.g. [45, 71,95]. Exceptions can be found. Robert and Kibler [104] proved that in special cases crackpropagation may occur without reducing the transverse stress intensity component to zero.The latter only takes place under special conditions and some authors therefore suggestbasing fatigue life predictions solely on mode I contribution, see e.g.[121, 150].

Some models, however, do take into account the mode II contribution. One way is byintroducing an equivalent stress intensity factor �KIeq in the Paris equation:

da

dN= C(�KIeq)

m (4.26)


The maximum stress criterion can be used to determine the equivalent mode I stress intensityfactor:

KIeq = KI cos3 �02� 3KII cos

2 �02sin

�02

(4.27)

where �0 denotes the direction in which the crack is most likely to propagate relative to thecrack tip coordinate system, and �KIeq is found as the KIeq range during one load cycle.

Tanaka [129] carried out experiments on cyclically loaded (R = 0.65) sheets of purealuminium with initial cracks inclined to the tensile axis. The original purpose of the studywas to establish threshold bounds for cyclic mixed mode fatigue crack growth. As a by-product, the experiments formed the basis for a crack propagation law:

da

dN= C(�Keq)

m

�Keq = (�K4I + 8�K4

II)1=4

(4.28)

where m was found to be 4.4 for the tested aluminium sheets. Eq. (4.28) was developed onthe assumption that a) the plastic deformation due to cyclic tension and transverse shearare not interactive, and b) the resulting displacement �eld is the sum of the displacementsfrom the two modes. Application of the law with satisfactory results can be found in [95].

The e�ect of the mean stress is assumed to be minor in the region of stable crack growth[51]. Nevertheless, Sih and Barthelemy [117] stated that a crack growth expression should,in principle, contain at least two loading parameters for a proper load de�nition. Thus, theyproposed to extend the strain energy density factor concept to cover the prediction of stablecrack propagation:

�a

�N= C(�Smin)

n (4.29)

where �Smin is given by

�Smin = a11(�0)[K2I max �K2

I min] (4.30)

+ 2a12(�0)[KI maxKII max �KI minKII min] + a22(�0)[K2IImax �K2

IImin]

where �0 is the angle of crack extension, a11, a12, and a22 are obtained by setting � equal to�0 in Eq. (4.18). The equation can be rewritten to include the stress intensity factor range�Kj and the mean stress intensity factors �Kj (for j = I,II):

�Smin = 2 [a11(�0)�KI�KI + a12(�0)(�KI

�KII +�KII�KI) (4.31)

+ a22(�0)�KII�KII]

4.6. Mixed Mode Crack Growth 55

Application of the approach can be found in [133], where C = 0.012 (cm/cycle)(kN/cm)�1:277

and n = 1.277 are used as material parameters for aluminium 2024-O. It was found thatthe estimated number of load cycles corresponded well to the values observed in the experi-ments. Badaliance [6] also related the crack growth rate to the range in strain energy density�Smin, but he used a relatively sophisticated model containing seven empirical constants forthe estimation of the crack growth.

Mageed and Pandey [78] compared experimentally obtained fatigue lives with fatiguelives estimated by both the strain energy density criterion and the equivalent stress intensityfactor methods given by Eq. (4.27). They found that Eq. (4.27) gave the best �t to theexperimental data with the best agreement for mode I dominating cases.

4.6.1 Threshold Range for Mixed Crack Propagation

Tanaka [129] was one of the �rst to investigate threshold limits for cyclic mixed mode loading.Specimens of pure aluminium were used for the analysis. Experimentally obtained thresholdvalues were compared to thresholds predicted by the maximum stress criterion and the strainenergy criterion, neglecting the e�ect of the peak stress intensity factors. The two criteriafor non-propagation were proposed to be:

cos�02

"�KI cos

2 �02� 3

2�KII sin �0

#= �Tc (4.32)

a11(�0)�K2I + 2a12(�0)�KI�KII + a22(�0)�K

2II = �Sc (4.33)

where �0 is the angle of crack propagation predicted by the two criteria. The quantities �Tcand �Sc are the mode I threshold values where �Sc can be found from

�Sc =1� 2�

4��(�KI;th)

2 (4.34)

When comparing the limits given by Eq. (4.32) and Eq. (4.33) with experimental results,Tanaka [129] found that Eq. (4.32) provides a conservative boundary for mixed mode loading,while a non-conservative boundary is predicted on the basis of the strain energy criterion.Tanaka made another �t to the experimental data by assuming that the condition for non-propagation is in a positive quadratic form of �KI and �KII:

A11�K2I + 2A12�KI�KII + A22�K

2II = 1:0 (4.35)


where the coe�cients for pure aluminium were determined as [129]: A11 = 0.0262 mm3=kg2,2A12 = 0.0081 mm3=kg2 and A22 = 0.0381 mm3=kg2.

The �t made by Tanaka [129] is based on a series of experiments with specimens madeof pure aluminium exposed to a load with the load ratio R equal to 0.65. Pook [93] usednineteen reported experimental threshold data sets for the generation of a more generalfailure mechanism map for mixed mode threshold behaviour. Theoretical upper and lowerbounds for the mixed mode threshold values were established. The lower boundary for stageII thresholds was expressed as a parabola:

�KII

�Kth

=

(0:08

��KI

�Kth

�2� 0:83

�KI

�Kth

+ 0:75

)1=2(4.36)

which is an appropriate limit for design purposes although it may be unduly conservative[93] in some circumstances.

4.6.2 E�ects of Biaxial Loading

Many engineering components are exposed to biaxial (multiaxial) static or cyclic loads.Biaxial or non-singular stresses acting parallel or perpendicularly to the crack plane canbe observed when a crack is propagating in a welding residual stress �eld. The e�ect ofbiaxial loading on crack propagation has been the subject of many investigations, see e.g.[57, 71, 118, 150]. Smith and Pascoe [118] made a literature review of experimentally observedbiaxial loading e�ect on fatigue crack growth. Unfortunately, no de�nite conclusion could bedrawn from the review. A cyclic tensile principal stress applied perpendicularly to the cracksurface was reported to increase, to have no in uence on, and to decrease the rate of crackpropagation when compared to uniaxial loading [118]. However, Hoshide and Tanaka [57]carried out experiments for di�erent stress ratios R and di�erent levels of the biaxiality ratio� (the ratio between the two non-singular stresses acting parallel or perpendicularly to thecrack plane). They eliminated the e�ect of biaxial loading by showing that the crack growthrate in biaxial stress �elds is a unique power function of �Ke� (except from cases wheresigni�cant plasticity is present). Their observation is supported by a later work of Kitagawaet al. [71], who also investigated cracks exposed to various biaxial loading conditions. Theyfound that the crack propagation curve including the threshold region can be characterisedsolely by �Ke�. All observations are restricted to small-scale yielding conditions. Whensigni�cant yielding is present biaxial loading may in uence the crack growth.

4.7 Limitations of the Linear Elastic Fracture Approach

Linear elastic fracture mechanics does not take into account the crack tip plasticity. Ductilematerials, like steel, always show some order of plasticity, which should be taken into con-sideration. However, in most cases, the linear approach (sometimes with modi�cations) is

4.8. Summary of Outlined Theory 57

su�cient to model high-cycle fatigue crack propagation of long cracks. The improvement ofan elastic plastic model applied to high-cycle fatigue is often very small according to Broek[23].

When the limitations of the linear elastic approach are evaluated, other possible modelapproximations and errors should be considered. Broek has been working on \sources of errorin fracture mechanics". He gives an overview [23] of the subject with a division of the errorsinto six main categories: 1) Intrinsic shortcomings of fracture mechanics, 2) uncertainty andassumptions as regards data input, 3) uncertainty due to aw assumptions, 4) interpretationsof and assumptions in stress history, 5) inaccuracies in stress intensity and, 6) intrinsicshortcomings of computer software. For long cracks in large structures the error due to linearelastic approaches is very small compared to the error due to data input and assumptionsin the stress history [23].

4.8 Summary of Outlined Theory

The purpose of this review is to establish a basis for predicting the behaviour of long crackssubjected to mixed mode loadings. A characteristic of mixed mode fatigue cracks is thatthey tend to propagate in a non-self-similar manner. Therefore, under mixed mode loadingit is not only the onset and the crack propagation rate which are of interest, but also thedirection of the crack propagation.

Three of the commonest methods for prediction of the onset of propagation and crackpropagation angles have been outlined and discussed. Information needed for determinationof the onset of brittle fracture is the fracture toughness of the material given either as Gcor as the critical stress intensity factor KIC. The outlined models for estimation of crackextension angles rely purely on the stress intensity factors at the crack tip which can bedetermined by �nite element procedures.

The crack propagation period is usually divided into three stages: a) Crack initiation, b)stable crack growth, and c) unstable crack growth where only the second stage is explicitlycovered by the present formulation. The initiation period is taken into account by assumingthat initial cracks exist in the structures of interest. This assumption is justi�able for weldedand ame-cut structures where cracks and aws are normally present at the initial stage dueto manufacturing procedures. After the initiation of mixed mode crack growth, the crackusually turns to propagate in the mode I dominating direction. Some authors thereforepropose to base the fatigue life estimation solely on mode I contributions. This has theadvantage that the material properties needed for the fatigue life predictions already existfor a wide range of materials. Another approach is to use an equivalent mixed mode stressintensity factor and then apply fatigue crack growth data from pure mode I tests. Thisprocedure will ensure conservative predictions of fatigue lives [95]. Glinka claims that themaximum e�ect of the KII contribution is in the order of 10 %.


The theory outlined in this chapter forms a solid basis for construction of a program forsimulation of mixed mode crack propagation. But some restriction has to be imposed on theformulation due to the complexity of the subject. No attempt has been made to cover theaspects of:

� Fatigue crack growth under variable amplitude loading.

� Out-of-phase mixed mode crack propagation.

� The e�ect of environment (corrosion, etc.).

� Material changes due to temperature, welding or cutting.

� The growth of short cracks (relative to the dimensions of the structure).

� Non-isotropic materials.

Two other important subjects have not yet been dealt with or they have only been brie ymentioned: The e�ect of crack closure which will be discussed in Chapter 6 and proceduresfor treating the e�ects of residual stresses which will be outlined in Chapter 7.

Chapter 5

Two-Dimensional Crack Propagation,

Simulation

5.1 Introduction

Finite element based methods have been successfully applied for solving fracture mechanicalproblems for several decades. The �rst applications were for relatively simple purposes suchas determination of stress intensity factors for di�erent crack con�gurations. However, thecomplexity of the models has increased signi�cantly over the years, as the e�ciency of thecomputers has increased. A state-of-the-art review of �nite element procedures used infatigue calculations, covering both linear elastic and plastic models, is given by Doughertyet al. [36]. The present work will focus on models for two-dimensional crack simulation.

Several procedures have been suggested for the prediction of crack growth in structureswhich are subjected to complex mixed mode stress conditions. Saouma and Zatz [110] weresome of the �rst to develop a model for automatic tracing of mixed mode crack growthin steel structures. They used a linear elastic quasi-static approach where the crack wasextended by a step-by-step scheme. The stress intensity state at each location of the cracktip was used to predict the direction of the next crack increment. The procedure involvedremeshing of the domain containing the continuously changing crack geometry and a modelfor prediction of the crack growth direction. The procedure was applied to the simulation ofcrack growth in an attachment lug and the obtained results showed a reasonable agreementwith experimentally obtained data. The weakness of the approach was the missing ability ofthe remeshing algorithm to produce a high quality mesh in the domain around the crack tip.Other researchers have worked on mixed mode crack simulation but basically it is the samestep-by-step scheme which is applied. Sumi [120, 121, 122, 123, 124] uses a combination of a�nite element and an analytical solution of the stress �eld at the crack tip to obtain both themixed mode stress intensity factors and the higher order stress �eld parameters. The higherorder terms allow for the use of curved crack increments based on a �rst order perturbation

59

60 Chapter 5. Two-Dimensional Crack Propagation, Simulation

Figure 5.1: Flow diagram and inputs for crack simulation procedure.

solution together with a local symmetry criterion. This model has with success been appliedto the estimation of crack paths and fatigue lives of complicated structures using a super-element technique [121]. Another type of multilevel substructuring technique was developedby Padovan and Guo [52, 90]. Displacements determined from a global level model were by atransformation imposed on the boundaries of the moving template containing the crack tip.A high mesh density in the template should then o�er an improved description of the stress�eld around the crack tip. The crack propagation in the global model was handled by the\element death method" where the elements in the wake of the crack are selectively killed.The template technique was tested for several examples including multicrack propagationand showed good correlation.

An objective of the present investigation is to analyse mixed mode crack propagation withspecial emphasis on the e�ects of residual stresses and biaxial loading. Di�erent designs ofcutouts in VLCCs are to be analysed for possible crack growth behaviours of cracks initiatedat points of high stress intensity. This chapter describes the basic components of the two-dimensional crack simulation procedure established for that purpose. An analysis of thecrack growth behaviour can reveal if crack arrest is more likely to occur in one design ratherthan in another. It should be mentioned that e�ects of residual stresses will �rst be discussedin Chapter 7.

5.2 Outline of the Simulation Procedure

The present simulation procedure is also based on incremental crack extension assumingthat the stress state in the vicinity of the crack tip can be described by a linear elastic

5.3. Stress Intensity Factors 61

fracture theory, which is usually justi�ed for high-cycle fatigue. The input to the procedureis a boundary de�nition of the initial structure with a crack located at a critical point. Thestructure can be subjected to both external and internal loads. The resulting stress andthe displacement �elds are analysed for each crack increment con�guration by means of the�nite element method. Special crack tip elements are used to describe the singular stressstate in the vicinity of the crack tip, and the stress intensity factors are computed by eitherthe displacement correlation technique or by use of a J-integral procedure. The directionof crack propagation is predicted from the obtained SIFs by one of the theories outlined inSection 4.3, and the crack is extended in the computed direction. The fatigue life can beupdated after each extension and local remeshing be carried out in the domain of the cracktip. The procedure is repeated until a stop criterion is reached (i.e. crack arrest, unstablecrack propagation, or a maximum number of simulation steps is reached).

The application of the theory outlined in the previous chapter will be brie y discussedin accordance with the ow of the procedure given in Figure 5.1. Special emphasis will beput on the numerical evaluation of stress intensity factors and the remeshing of the domainaround the crack tip. The capability of the procedure will be demonstrated by some examplesof crack simulations.

5.3 Stress Intensity Factors

The stress intensity factors are the governing parameters in a linear elastic approach anda large number of �nite element based procedures have been established for the purpose ofevaluating the stress intensity factors. In the early studies, a high-density mesh was needednear the crack tip for the singular stress �eld. Later on, hybrid crack tip elements wereintroduced which contained the theoretical singularity of the stress �eld (an early work waspresented by Byskov [24]). These elements lacked the capability of the constant strain andthe rigid body motion modes [15] and they therefore did not pass the patch test [30], whichis a necessary requirement for convergence. Barsoum [14, 15] and Henshell and Shaw [55]have independently shown that the quarter-point isoparametric elements allow for a highlyaccurate computation of the stress �eld with a relatively coarse mesh around the crack tip.These singular quarter-point elements provide a good tool for both linear and non-linearfracture mechanics calculations.

5.3.1 Singular Isoparametric Quarter-Point Elements

The commonly used singular elements for two-dimensional calculations are: 1) 8-nodedquadratic isoparametric elements, where the singularity is achieved by placing the mid-sidenode near the crack tip at the quarter-point, or 2) 8-noded triangular elements obtained bycollapsing one side of the quadratic elements and placing the mid-side node at the quarter-point (the two types of elements are illustrated in Figure 5.2). The latter method leads


Figure 5.2: Collapsed quarter-point elements.

to far the best results, which is due to the fact that the square-root singularity is obtainedthroughout the triangular element, whereas for the quadrilateral elements, this behaviour hasbeen shown to exist only along the side of the elements [13]. Thus, the triangular collapsedelements were chosen for the present formulation.

5.3.2 The Displacement Correlation Technique

The displacement correlation technique proposed by Shih et al. [114] is a very simple methodfor calculating mode I stress intensity factors by use of the displacements of the singularquarter-point elements. The expression for the determination of KI was obtained by com-bining Eq. (4.2) and the singular part of the displacements along the crack face (Appendix Agives an expression for the displacement along the edge of a quarter-point element). Ingraf-fea and Manu [61] expanded this approach to a mixed mode condition where the equationsread:

KI =2�

�+ 1

r�

2h[4u2,B2 � u2,C2 � 4u2,B1 + u2,C1]

KII =2�

�+ 1

r�

2h[4u1,B2 � u1,C2 � 4u1,B1 + u1,C1]

(5.1)

here � is the shear modulus of the material, and � is given by Eq. (4.3). The quantitiesu1 and u2 are the sliding and opening displacements of the crack tip element at the pointsdepicted in Figure 5.3 and h is the length of the side of the element which is common tocrack face.

5.3.3 Stress Intensity Factors by the J-Integral

The rate of change of total potential energy relative to the change in crack length can becalculated by the J-integral. It has been shown that this integral is valid from small scale


Figure 5.3: Location of nodes in local coordinate system.

yield to a state general of yielding and that it is equal to the energy release rate underlinear elastic conditions. The critical value of the J-integral can be considered as a fracturemechanical material property. The stress intensity factors, the energy release rate and theJ-integral are in the two-dimensional case related by the following non-linear equations [38]:

J1 = G1 = K2I +K2

II

E 0

J2 = G2 = �2KIKII

E 0(5.2)

where J and G are components in the global coordinate system. The parameter E 0 is equal toYoungs modulus E for a plane stress condition and to E=(1��2) for a plane strain conditionwhere � is Poissons ratio.

5.3.4 Procedures for J Computations

The J-integral can be computed by two di�erent approaches 1) as a contour integral goingfrom the lower to the upper crack surface, or 2) by a virtual crack extension method. Bothtechniques are summarised in this section, where the latter method will be described in theform of an equivalent domain-integral.

Contour Integral Method

The path-independent J-integral, originally proposed by Rice and Rosengren [102], is aclosed contour integral of the strain energy density and the work done by traction aroundthe crack tip. The integral was derived on the assumptions that: a) The body consists ofa homogeneous material, b) the body is subjected to a two dimensional stress �eld, c) the


Figure 5.4: Crack-tip contours. Figure 5.5: Integration paths.

crack surface is tension-free, and d) the body forces are zero [101]. Generally, the contourintegral assumes the form:

Jk = lim�!0

Z��

"Wnk � �ij

@ui@xk

nj

#d� k = 1; 2 (5.3)

where W denotes the strain energy density of the material, and n is a normal vector pointingout into the surrounding domain as de�ned in Figure 5.4. The body is divided into elementswhen the �nite element method is applied and the contour integral is simply calculated asthe sum of the contributions from each element intersected by path �. The path across theelement is free in principle, but Bakker [9] strongly suggests that paths are restricted torunning through the element integration points to avoid extra interpolation for the deter-mination of stresses and strains within the element. It is also recommended to use a pathrunning parallel to the local element coordinates as illustrated in Figure 5.5 and then applynormal Gaussian integration procedures.

Numerical Procedure for Contour Integral Method

Choosing an integration path with the local element coordinate � corresponding to either �1or �2 as suggested by Bakker, the contour integral maps into a summation through Gaussianpoints of the elements intersected by the contour:

Jc;x =Xe

NoXn=1

HnIce(�) (5.4)

where e applies to the elements intersected by the contour �. No is the order of integrationfor the element and Hn is the weight factor at the location �. The integrand of Eq. (5.4) isin the local element coordinate system given as

Ice(�) = (Wnk � �ijui;knj)ds

d�k = 1; 2 (5.5)


Figure 5.6: Contours for the EDI method.

here s is the path length in the global coordinates. The normal vector components timesthe derivative of the path length with respect to the local element coordinate can be foundfrom

n1ds

d�=

dx2d�

=PXp=1

@NP

@�xp2

n2ds

d�= �dx1

d�= �

PXp=1

@NP

@�xp1

(5.6)

where P is the number of element nodal points and NP (�1; �2) are the nodal point shapefunctions at points (�1; �2).

Equivalent Domain Integral Method (EDI)

The equivalent domain integral method (EDI) is an alternative way to obtain the J-integral.The contour integral is replaced by an integral over a �nite-size domain. The EDI approachhas the advantage that the e�ect of body forces can very easily be included.

The standard J-contour integral given by Eq. (5.3) is rewritten by the introduction of aweight function q(x1; x2) [87]:

Jk =Z�

"Wnk � �ij

@ui@xk

nj

#qd� k = 1; 2 (5.7)

The weight function assumes unit value at the inner contour C1 and zero at the outer contourC2 (the contours are depicted in Figure 5.6). The introduction of the q-weight function can


be interpreted as a virtual crack extension technique by shrinking the inner contour C1 tothe crack tip. The material at the tip is thus given a virtual displacement of one, whilethe surrounding material at the C2 contour is �xed. The weight function may assume anarbitrary value between the two contours describing the area AI . A practical choice wouldbe a linear interpolation between the values at the boundary contours.

The line integral given by Eq. (5.7) can for a traction-free crack surface be transformedinto a domain integral by means of the divergence theorem. Thus, the equivalent domainintegral is derived. It reads:

Jk = �ZA

(W

@q

@xk� �ij

@ui@xk

@q

@xj

)dA�

ZA

(@W

@xk� @

@xj

"�ij

@ui@xk

#)qdA k = 1; 2 (5.8)

The second term of Eq. (5.8) vanishes for structures of homogeneous materials in the absenceof body forces. A major advantage is that the integral formulation from Eq. (5.8) is relativelyinsensitive to the mesh con�guration, unlike other methods, e.g. Irwins crack closure integral[28]. Problems might, however, occur in the evaluation of J2 when the non-singular termsnear the crack tip become dominant. A way of avoiding this di�culty is described in thefollowing section.

Improving the EDI Method

The domain integral given by Eq. (5.8) is valid for evaluation of Jk when the singular stressesdominate the stress �eld but for general mixed mode problems, the stress distribution nearthe crack tip sometimes contains both singular and non-singular terms. The area integral isderived on the assumption that there is no contribution to Jk from the line integral along thecrack surface (�+s + ��s depicted in Figure 5.4). This holds for the calculation of J1 but inthe calculation of the J2, there are terms which become non-zero if non-singular stresses arepresent [98]. Therefore, the line integral along the crack surface must be included if generaland accurate results are to be obtained for the J2-component. This is unfortunate since allthe parameters needed for the numerical integration are known in the Gauss points whichare not located along the crack surfaces.

One way of avoiding the line integrals is to apply the decomposition method [28, 98, 112].By this approach, the displacements and stresses are decomposed into symmetric (mode I)and antisymmetric (mode II) �elds with respect to the crack tip. Raju and Shivakumar[98] have shown that the contribution from the line integrals along crack faces vanish in thedecomposed stress �elds.

Consider two points P (x1; x2) and P 0(x1;�x2) in the vicinity of the crack tip which aresymmetrically located with respect to the crack line (see Figure 5.7). Displacements and


Figure 5.7: De�nition of points for symmetric and antisymmetric decomposition.

stresses at these points can, according to Sha and Yang [112], be used for an analyticalseparation into the two following fracture modes:

fug = fuIg+ fuIIg = 1

2

(u1 + u01u2 � u02

)+1

2

(u1 � u01u2 + u02

)(5.9)

and

f�g = f�Ig+ f�IIg = 1

2

8><>:

�11 + �011�22 + �022�12 � �012

9>=>;+

1

2

8><>:

�11 � �011�22 � �022�12 + �012

9>=>; (5.10)

A similar decomposition can be made for the strain �eld. By decomposing the computedstress, strain and deformation �elds into mode I and II components, the J-integral may beevaluated as [28]:

JI1 = J1 =ZA

(�Iij

@uIi@x1

@q

@xj�W I @q

@x1

)dA+

ZA

(@

@xj

"�Iij

@uIi@x1

#� @W I

@x1

)qdA (5.11)

JII1 = J2 =ZA

(�IIij

@uIIi@x1

@q

@xj�W II @q

@x1

)dA+

ZA

(@

@xj

"�IIij

@uIIi@x1

#� @W II

@x1

)qdA

Note that it is the J�1 component (k = 1 in Eq. (5.8)) which is used for the calculation. TheJ�2 components will be identically zero because of the symmetric conditions, thus there isno contribution from the line integral along the crack surface. Obviously, the decompositionmethod involves one more step to calculate the new stress, strain and deformation �elds, buthigher accuracy is obtained and the individual modes of the J-integral are directly available.


Numerical Procedure for Equivalent Domain Integral

When the structure is of homogeneous material and the body forces are absent, the �niteelement implementation of Eq. (5.11) becomes very similar to the implementation of thecontour integral. The only real di�erence is the introduction of the weight function q. Withthe isoparametric �nite element formulation the distribution of q within the elements can bedetermined by a standard interpolation scheme by use of the shape functions N :

q =8Xi=1

NiQi (5.12)

where Qi are values of the weight function at the nodal points. The spatial derivativesof q can be found by use of the normal procedures for isoparametric elements. When thespatial derivatives of the weight function are known, the equivalent domain integral can becalculated as a summation from the discretised form of Eq. (5.11):

J�1,EDI =X

elements

in AI

PXp=1

" ��ij

@u�i@x1

�W ��1j

!@q

@xjdet

@xm@�m

!#p

wp � = I; II (5.13)

The terms within [ � ]p are evaluated at the Gaussian points P , where wp is the Gaussianweight factor for each point. The present formulation is for a structure of homogeneousmaterial in which no body forces are present.

5.3.5 Evaluation of Numerical Methods for Determination of SIFs

Stress intensity factors obtained by the presented numerical techniques were compared tosome benchmark examples. Four cases are given in Appendix B covering pure mode I andmixed mode conditions. A good agreement was obtained for the determination of both themode I and mode II stress intensity factors. This section focuses on a closed-form exampleconsisting of a biaxially loaded large plate with an inclined crack at the centre. If �nite-sizee�ects are neglected, then an analytical expression for stress intensity at the crack tip canbe found for the con�guration depicted in Figure 5.8. It reads:

KI =��22 sin

2 � + �11 cos2 ��p

�a

KII = (�22 � �11) sin� cos �p�a

(5.14)

Numerical calculations were performed for uniaxially loaded plates (�11 = 0) with crackshaving di�erent inclination angles. The length h and the width w were taken to be 20 times

5.4. Crack Arrest Criterion 69

Figure 5.8: Centre crack con�guration. Figure 5.9: Obtained results for centre-crackedspecimen.

the half crack length a (trying to minimise the �nite-size e�ect). This con�guration was usedto test the determination of stress intensity factors by the displacement correlation techniqueand the J-integral in the formulation of the decomposition method. Figure 5.9 shows modeI and mode II stress intensity factors normalised by KI at � = 0 as a function of inclinationangle. From the �gure it can be seen that the two methods provide accurate results for themode I factors and that the simple displacement correlation technique also produces quiteaccurate results for mode II stress intensity factors. Unfortunately, quite the same trend isnot observed for the J-integral techniques, where there is an error in the determination ofthe KII factor (in the worst case for � = 300, the EDI method shows a 7 % deviation).

Crack propagation angles �0, based on SIFs calculated for � = 300, were determinedby the MTS criterion (see Eq. (4.11)) to evaluate how the inaccuracies in the numericaldetermination of KII in uence the crack path simulation. The theoretical angle of crackpropagation for � equal to 300 is 43:220. The crack propagation angles obtained by thedisplacement correlation technique and the J-EDI method are 42:840 and 45:080, respectively.The maximum deviation from the theoretical angle is about 4 % for the EDI method, whilethe displacement correlation technique shows excellent agreement with a deviation of lessthan 1 % from the theoretical angle. On the basis of these results and the examples presentedin Appendix B it is concluded that both DCT and EDI techniques provide mixed mode stressintensity factors with an accuracy suitable for crack path simulation.

5.4 Crack Arrest Criterion

When crack arrest is discussed a distinction must be made between static and cyclic crackgrowth. For the static mixed mode case, crack arrest can be determined from one of thecrack extension criteria outlined in Section 4.3.1.


Crack arrest for cyclic crack growth is related to the threshold value �Kth, which isde�ned as the limit below which the crack remains dormant or propagates at a non-detectablecrack growth rate. The crack arrest criterion for cyclic loading is therefore taken to be whenthe stress intensity at the crack tip falls below the mixed mode fatigue threshold limit.

5.5 Extension of the Crack

The crack is extended if crack arrest has not been detected. Three criteria are availablefor the determination of the crack propagation direction: 1) Maximum tangential stress, 2)minimum strain density, and 3) maximum energy release rate, all discussed in Section 4.3.Given the angle of propagation, the new location of the crack tip is solely determined bythe size of the crack increment, which is a user input in the present formulation. The choiceof an appropriate crack increment length is governed by two contradictory requirements, 1)accuracy and 2) simulation time. In situations where relatively high ratios of KII /KI areexpected or in regions with high stress intensity factor gradients, it is proposed to reducethe increment size to improve the accuracy.

5.6 Updating of Fatigue Life Estimation

The estimation of fatigue life can be updated for each crack extension. The crack growthequation provides a relation between the crack increment �a and the increment in thenumber of load cycles �N . The number of load cycles equivalent to the crack increment canfor cyclically loaded structures be determined by a numerical integration of the governingcrack growth equation. If the contribution from the unstable crack growth is neglected, thenmodi�cation of Eq. (4.24) can be used for the integration, taking into account the e�ect ofthe threshold region:

NIncrement =Z ar+�a

ar

da

C[�Kmeq ��Km

eq;th](5.15)

where C and m are material parameters, Keq is the equivalent mode I stress intensity factor(discussed in Section 4.6), and Keq;th is the corresponding threshold value. The crack lengthis equal to ar before it is extended by the increment �a.

5.7. Remeshing of the Crack Domain by the Advancing-Front Method 71

5.7 Remeshing of the Crack Domain by the Advancing-

Front Method

5.7.1 Introduction

When a step-by-step procedure is applied to automatic tracing of non-collinear fatigue crackpropagation, the continuously changing crack path requires local remeshing in the zonecovering the crack in each step of a simulation. To avoid user interference, the meshingprocedure must be automatic and very robust. It has to meet the normal requirements suchas precise modelling of boundaries, applicability to general topologies, and it has to createelement shapes suitable for �nite element analysis. Often a transition from small elementsin the vicinity of the crack tip to larger elements at the boundaries is necessary in the caseof crack path simulation. Hence, the mesh procedure also has to perform well under suchconditions.

In the present formulation, the crack propagation is assumed to take place in a two-dimensional plane. Many approaches are available for plane mesh generation, covering bothtriangular [7, 33, 48, 63, 64, 73, 94] and quadrilateral [7, 65, 94, 136] element generation.When the mesh generation algorithm is to be chosen the following items must be consid-ered: 1) The robustness of the method, 2) the cost of implementation (simple or complexalgorithm), 3) the speed of the generation, and 4) the accuracy of the chosen element typerelative to the degrees of freedom required for the modelling.

Figure 5.10: The active and the passive remeshing zones.

The zone in which the crack is likely to propagate is often relatively small compared tothe rest of the structure. This is illustrated by the grey active zone relative to the whitepassive zone in Figure 5.10. Consequently, only a small part of the structure has to beremeshed in each step of the crack propagation. It is therefore the robustness and the costof implementation which are of major concern. Among the various techniques for generatingunstructured meshes in domains of arbitrary geometries, the advancing front method hasproved to be very successful [48, 63, 64]. This technique is therefore chosen for the continuousremeshing of the active zone around the crack tip. In the next section an outline of themethod is given and in Section 5.7.9 the application to simulation of crack propagation isdescribed.


Figure 5.11: Flow for the mesh generation pro-cedure.

Figure 5.12: Input for the mesh gen-eration.

5.7.2 Meshing Strategy for the Advancing-Front Method

The mesh is constructed from the boundaries when the advancing-front mesh generationprocedure is applied. It can be implemented by two di�erent approaches: (1) all the interiornodes are generated before the construction of the triangular elements [69, 73], or (2) theinterior nodes and triangular elements are constructed simultaneously [48, 64]. The latterapproach seems to be the most attractive method since it allows for some degree of controlof the shape of each generated element [33]. This approach is employed in the presentformulation and the ow of the implementation is sketched in Figure 5.11. The descriptionof the triangulation procedure will follow this sketch.

5.7.3 Initialising the Front

The �rst step of the triangulation procedure is to generate the initial front face list fromone or more closed loops of straight-line segments. Basically, the front face list contains thediscretised boundary of the domain to be meshed. Each face represents a potential side fora new element. The face list is constructed from an input �le containing information aboutthe boundary segments, the distribution of the nodes along the segments, and the numberof nodes on each segment. The segments are de�ned by a starting and an ending point(the control points are depicted in Figure 5.12), speci�ed counter-clockwise for the externalboundary edges and clockwise for the inner loops which are also depicted in Figure 5.12.


Figure 5.13: Flow in element generation procedure.

This de�nition of the boundaries allows for element generation in domains containing bothinternal and edge cracks.

5.7.4 Departure Zone

The departure zone is the selected front face segment from which the new element is gen-erated. Di�erent criteria can be used in the determination of the departure zone, but it isstill not clear how to de�ne an optimal strategy which covers all geometrical cases [48]. Thesmallest face segment of the active front was successfully chosen as the departure zone inthe present formulation.

5.7.5 Make New Element

Creating a new element is the heart of the mesh generation. A sketch of this central partis given in Figure 5.13. After the departure zone has been selected, the main goal is to �ndthe best location for the new third node.


Figure 5.14: De�nition of control circle. Figure 5.15: Merging of two widely dif-ferent fronts.

Potential Third Node Location

A new element is created by use of the departure zone as a basis (line AB in Figure 5.14).The �rst step is to determine a potential third node location (PTNL). The third node(point C in Figure 5.14) may be constructed so that the triangle ABC is equilateral, whichensures elements of a constant size which are well shaped for �nite element calculations.Unfortunately, this method may cause trouble when a front with small elements is goingto approach a front with relatively large elements since the created equilateral elements areof constant size (see Figure 5.15). A way to avoid this problem is to create non-equilateralelements allowing them to grow in size. Problems might still occur if the elements have togrow from a small size to a very large size over a short distance. The mesh density can beforced to grow rapidly in size by applying a control space to govern the size of the elements.At the initial stage, the control space de�nes a desired size of the elements (and thus theinitial element height) in all of the domain which is to be meshed (for details see [48]). In thecase of rapid element growth, however, elements of bad shapes will unavoidably be generatedwith large di�erences between the length of the three sides of the element. Rapid elementgrowth is almost always a consequence of poor initial boundary discretisation, so properinitial boundary de�nitions are the best way to ensure a successful mesh generation.

An Existing Node in the Vicinity of a PTNL

The second step is to decide if an already existing node can be used for the generation of thenew element. If the PTNL is too close to an existing node, then it is of importance that thenew element \snap" to the existing node. This is ensured by testing if an existing node closeto a potential third node location is within a distance de�ned by a control circle (the controlcircle is illustrated in Figure 5.14). A node found inside the control circle becomes the newPTNL. It is the size of the control circle and the initial location of the third node whichcontrol the growth in element size, since a large control circle rather often causes snappingto an existing node.


Figure 5.16: The three checks carried out for the third node.

Checking if the PTNL is Quali�ed for Element Generation

The third step is to analyse the chosen location of the PTNL (point C in Figure 5.16). Thisanalysis consists of three checks:

1. Is the potential third node location inside an existing element?

2. Is the potential third node location too close to an existing front face?

3. Do the two faces AC and BC cross or overlap one of the existing faces?

The three checks listed above are illustrated in Figure 5.16. The order of the checks isimportant, since the �rst check is faster to carry out than the third check and the locationof the node is not quali�ed if just one of the checks fails. The �rst and the third checksshould guarantee that a proper element is generated at the present stage, while the secondcheck is included to ensure that the remaining zone can be dealt with without encounteringdi�culties at a later stage, that is ill-shaped elements are sought to be avoided. If one ofthe checks fails, then the \check procedure" suggests an existing node in the vicinity of thepresent PTNL as the new third node location (position D or E in Figure 5.16). The newthird node location is analysed and the procedure is continued until a suitable node positionis found and �nally the element is generated.

5.7.6 Update Front/Front Empty ?

The front is updated after the generation of the new element. Updating the face list means:1) Delete the face which was used as a basis for generating the new element, 2) add faces ofthe element just formed which are not common to any existing elements, and 3) in the caseswhere an existing node has been selected as the third node location, delete the faces whichare common to the newly generated element and the existing elements. After updating thefront, it is checked if the front is empty. In that case, the method has converged and themeshing of the domain is completed.


Figure 5.17: The used smoothing procedure.

5.7.7 Smoothing of the Generated Mesh

The smoothing process follows upon a complete triangulation. All nodes at the externalboundaries are �xed, but the location of the interior nodes is shifted by a smoothing proce-dure. Here, a relatively simple smoothing procedure is employed. The new location of thenode P 0 is determined as the average value of the centre of gravity of the n elements sharingthe node P (see Figure 5.17):

P 0 =1

n

nXj=1

Gj (5.16)

where Gj is the centre of the n elements. Smoothing is an iterative procedure and it isrepeated until there is no signi�cant change in the nodal position between iterations. Thisusually happens after a few iterations (�ve iterations are used in the present implementation).Other smoothing procedures are available, such as barycentrage and relaxation [48], but nomatter what procedure is used, it has to be ensured that the quality of the mesh remainsvalid after the smoothing.

5.7.8 Improvement of the Method

The advancing-front method technique requires an extensive search of nodes and faces dur-ing the generation of new elements. Data structures, such as Quadtree [7, 48] could beimplemented to speed up the search routines. However, in the present case the remeshing isrestricted to a relatively small region and attempts to improve the speed of the generationare therefore not found to be necessary.

5.7.9 Application to Cracked Structures

Poorly shaped elements may be generated at the crack tip if the meshing algorithm is applieddirectly to a crack propagation problem [94]. Thus, a template is used at the crack tip to


Figure 5.18: An illustration of a template con�guration where the crack front is indicatedby the dots.

guarantee well shaped elements around the crack tip, and to ensure that the mesh geometryis the same in the vicinity of the crack tip for each step of the crack propagation simulation.The outer geometry of the template is �xed, so that the elements within the template (theelements in the shaded area in Figure 5.18) are not a�ected by the smoothing procedure.Inside the template, quadrilateral elements and collapsed quadrilateral elements are used forthe meshing. The mesh density in the template area is determined by a re�nement parameterwhich denotes the number of subdivisions. Thus, it is easy to check the convergence ofdi�erent parameters (i.e. the stress intensity factors) as a function of an increasing meshdensity around the crack tip.

5.7.10 Examples of Mesh Generation

The proposed mesh generation algorithmwas implemented in a C-program and the procedureturned out to be very stable, fast, and robust. Three examples are given here to demonstratesome of the capabilities of the mesh generator. The two �rst examples depicted in Figure5.19 and Figure 5.20 show that the generation algorithm is capable of handling domainswhich contain both convex and non-convex holes, and that the algorithm performs well incases of transition from relatively small element sizes to large element sizes. Figure 5.20also shows that it is possible to concentrate the mesh density in some regions (by the initialboundary de�nitions).

After the initial test the mesh generation algorithm was successfully implemented in thestep-by-step based crack simulation program. Curved crack paths can be found when thecrack propagation is in uenced by geometrical discontinuities such as holes. Simulationswere performed for a crack propagating in a simply supported beam with three holes. Thebeam was subjected to a single-force load at the centre of the beam and Figure 5.21 showsthe mesh for the second step in a simulation. It is seen that the holes are nicely describedand that the mesh density is concentrated in the zone around the crack tip. The �gure alsoindicates that the right side of the structure is modelled by a static mesh which does notchange during the crack propagation.


Figure 5.19: Domain containing both convex andnon-convex holes.

Figure 5.20: Merging of two widely dif-ferent fronts.

Figure 5.21: Example of mesh generation.

5.8. Examples of Crack Path Simulation 79

5.8 Examples of Crack Path Simulation

Crack paths obtained by the simulation procedure are in this section compared with nu-merically and experimentally obtained results which cover the in uence of holes and biaxialloading.

5.8.1 In uence of Holes on the Crack Growth

It has been found both experimentally and theoretically that the presence of holes may havea signi�cant in uence on a propagating crack and holes have even been suggested as crackarresters [123]. Bittencourt et al. [20] described some experiments carried out with staticallyloaded beams containing holes. The beams were made of the polymer PMMA, which is acommon choice of material for crack path investigations, since it is relatively homogeneousand exhibits brittle fracture behaviour at room temperature. Experiments were made forthe two di�erent initial crack con�gurations called type I and II in Figure 5.22. Crack paths

Figure 5.22: Initial geometry of specimens used in the experiments.

were predicted by the simulation procedure, using the equivalent domain integral methodfor determination of the stress intensity factors and the maximum tangential stress criterionfor calculation of the crack propagation angle. The experimental and the simulated cracktrajectories are compared in Figure 5.23 for both the initial crack con�gurations (there isonly one crack at a time in the simulations and the experiments). A good agreement isfound between the simulated and the experimentally determined crack paths for the type IIcon�guration. The same tendency is found for the type I specimens. However, a deviationis observed when the crack tip approaches the hole. The simulation fails to predict the turnof the crack into the hole. The experimentally observed turn of the crack is believed to berelated to plastic deformations of the material and it is therefore not predicted by the linearelastic approach.

The type II con�guration was used to test the di�erence in simulated crack trajectoriesby application of respectively the J-EDI and the DCT method for calculation of stress


Figure 5.23: The experimental and the simulated crack trajectories depicted in the same�gure for the two con�gurations.

Figure 5.24: Simulated crack paths by use ofthe J-EDI and the DCT method for determi-nation of stress intensity factors. No visualdi�erence.

Figure 5.25: Simulated crack paths by useof the three di�erent criteria for determina-tion of crack propagation angle. No visualdi�erence.

5.8. Examples of Crack Path Simulation 81

Figure 5.26: Experimental and simulated crack paths in a non-symmetrical specimen exposedto a lateral force. The initial mesh for the simulation is also depicted.

intensity factors. Figure 5.24 shows the two predicted crack paths, which were found almostto coincide, by use of the maximum tangential stress criterion for determination of crackpropagation angles. The same con�guration was also employed in the test of the threemethods outlined in Section 4.3.1 for estimation of the crack propagation angle. Figure 5.25shows the obtained crack paths where the DCT was used for calculation of the stress intensityfactors. From the two �gures, it is concluded that the choice of procedures for determinationof SIFs and crack propagation angle does not a�ect the simulation signi�cantly.

5.8.2 Non-Symmetric Specimen Exposed to Lateral Force Bend-ing

The experiments described by Bittencourt et al. [20] were based on the brittle polymerPMMA. Experimental measurements of crack propagating in structural steel (E = 2.1�1011Pa and � = 0.3) were made by Theilig et al. [138], using a non-symmetric specimen exposedto a bending moment and a lateral force. A sketch of the specimen is given in Figure 5.26,where the initial mesh for the present simulation is also depicted. Notches were attachedalong the circular region of the specimens de�ned by the angle �. Experiments were carriedout for three di�erent positions of the initial crack and �ve specimens were investigated foreach initial position of the crack. The experimental scatter band was, according to Theilig etal. [138], found to be relatively narrow which can also be seen from the �gure. The structurewith the thickness t equal to 15 mm was subjected to a lateral force FQ of the magnitude 1kN and a bending moment Mb equal to 200 Nm. The experimental scatter bands and thesimulated crack trajectories are depicted in Figure 5.26 for � equal to 400. It is seen thatthe simulation provides a good description of the experimentally obtained data. Theilig et


al. [137] used the same experimental con�guration for veri�cation of a simulation programbased on a higher order method called the MVCCI method (for details, see [137]). They alsoobtained a �ne representation of the crack trajectories.

5.8.3 Biaxial Loading

From failures observed in practice, it is known that cracks are frequently initiated in regionscharacterised by complicated geometrical shapes which are asymmetrically loaded [137].These conditions may result in curved crack trajectories and cracks initiated in membersof secondary structural importance can thus propagate into primary members which arevital to the structural integrity. To illustrate the problem, Sumi et al. [124] carried outnumerical crack path simulations with biaxially loaded cruciform joints. The geometry ofthe specimens is depicted in Figure 5.27 with the biaxial stress range r de�ned as the ratioof the maximum bending stress range applied at the end of the joint. Figure 5.27 alsoshows the crack paths obtained by Sumi et al. [124]. The simulation procedure discussedin this chapter was applied to the same con�gurations and the predicted crack trajectoriesare presented in Figure 5.28. There is a good agreement between the crack paths obtainedby the two di�erent approaches. Both procedures predict that small biaxial stress rangesr can result in that a crack initiated in a secondary component propagates into a primarycomponent due to the biaxial loading (which may cause failure of the whole structure).

5.9 Summary

A step-by-step crack simulation procedure has been presented. The three commonest crackextension criteria were employed. The di�erent components of the procedure were discussedwith special focus on the numerical evaluation of the stress intensity factors. In the presentformulation these can be determined by both the displacement correlation technique and theJ-integral in the equivalent domain integral formulation with decomposition applied. Cracksimulation showed that the choice of methods for determination of crack propagation angleand stress intensity factors does not in uence signi�cantly the prediction of the crack paths.

A simple and robust procedure for automatically triangular mesh generation in planedomains of arbitrary geometries has been described. Special attention has been paid to thegeneration of meshes in the domain surrounding a propagating crack. This implies thatthe procedure is able of to handle cases where a transition is necessary between regions ofsmall elements found in the vicinity of the crack tip and relatively large elements located atthe boundaries of the domain. A template was employed at the crack tip to guarantee wellshaped elements around the crack tip and to ensure that the mesh geometry is the same inthe vicinity of the crack tip for each step of the crack propagation simulation. The procedurehas proved to be well suited to support of discrete crack propagation.

5.9. Summary 83

Figure 5.27: Crack paths simulated by Sumi [124] for a cruciform joint under biaxial in-planebending.

Figure 5.28: Crack paths simulated by the present simulation procedure.


The simulation procedure was validated by comparison with experimental and numericaldata covering the in uence of both holes and biaxial loading. It is therefore concluded thatthe presented crack simulation scheme is able to handle crack propagation in general twodimensional domains.

Chapter 6

Crack Closure

6.1 Introduction

Contact between the two crack surfaces during a fatigue load cycle is called crack closure.Several conditions may lead to crack closure. The two most important are: I) Plastic-inducedclosure caused by the plastic deformed material left in the wake of a propagating crack. Thephenomenon has been extensively discussed in the literature since it was �rst described byElber [39]. He found that the plastic wake can cause closure of the crack surface even during afully tensile load cycle. A second source of plastic-induced closure is overloads. An overloadmay create a plastic region in front of the crack tip. A compressive residual stress zoneforms during unloading, which results in closure at the crack tip. Plastic-induced closure isbelieved to be the main source of sequence e�ects which can a�ect signi�cantly the crackgrowth under variable amplitude loading (sequence e�ects are brie y discussed in Section4.5.3). II) Geometrical closure arising from a combination of specimen geometry and loadson the structure such as external bending and compressive residual stresses. This type ofclosure is related to the mean stress at the crack tip, often described by the stress ratio R(R = �min=�max or R = Kmin=Kmax).

Oxide- and roughness-induced closures are two minor important mechanisms which areonly interesting when the stress intensity range at the crack tip approaches the thresholdlevel. A corrosive environment may lead to oxidation of newly cleft crack surfaces. Oxidedebris on the crack surfaces then cause a wedging e�ect imposing a rise in the stress intensitylevel at which the crack closes. Roughness-induced closure is related to low stress intensitylevels at which the dimensions of the plastic zone at the crack are smaller than those ofthe characteristic microstructure such as grain size. It is therefore highly metallurgical andmicrostructure-related. The mechanisms of the two latter types of closure are illustrated inFigure 6.1.

A direct analysis of plastic-induced closure is not within the framework of linear elasticfracture mechanics and it is therefore beyond the scope of this formulation. The e�ect of

85

86 Chapter 6. Crack Closure

Figure 6.1: A schematic illustration of oxide- and roughness induced crack closure.

plasticity will therefore only be indirectly included by the semi-empirical formulas whichrelate the linear elastic stress state at the crack tip to the crack growth. Attempts have beenmade to model both oxide-induced closure [27] and roughness-induced closure [10, 84, 141].However, these models are speci�ed by a high number of uncertainties and assumptions, e.g.the actual surface topography, including the location of oxide debris, and crack surfaces areassumed to be in�nitely hard. Hence, the e�ect of oxide- and roughness-induced closure isalso only indirectly taken into account by a semi-empirical description of the crack growth.

Crack closure, whatever its origin, a�ects the stress intensity at the crack tip and thusalso the crack growth. The damaging part of the load cycle is reduced by the closure of thesurfaces. When linear elastic fracture mechanics is applied, the reduction is usually handledby the introduction of an e�ective stress intensity range. The aspect of an e�ective stressintensity range is outlined in the next section, while a �nite element approach which can beused for the modelling of geometrical crack closure is discussed in Section 6.3.

6.2 E�ective Stress Intensity Range

It is normally assumed that mode I cracks are unable to propagate while they remain closed,and the net e�ect of closure is a reduction of the nominal stress intensity range to a lowervalue �Ke� called the e�ective stress intensity range. �Ke� is the portion of the fatiguecycle for which the crack is fully open at the crack tip [89]. The e�ective stress intensityrange can be determined as

�Ke� = Kmax �Kcl for Kmin � Kcl

and

�Ke� = Kmax �Kmin for Kmin > Kcl

(6.1)

where Kmax and Kmin are the maximum and the minimum stress intensity levels appliedduring a fatigue load cycle. The closure stress intensity Kcl is, in the formulation of Packiaraj

6.2. E�ective Stress Intensity Range 87

Figure 6.2: De�nition of di�erent stress intensity levels.

et al. [89], de�ned as the maximum stress intensity at which the crack surfaces are rigidlypressed together and remain closed during unloading. Sometimes the crack closure intensityin Eq. (6.1) is replaced by the opening stress intensity Kopen, de�ned as the minimum stressintensity at which the crack tip is fully open and the compressive contact stresses behindthe crack tip are overcome during the loading phase [89]. The di�erent stress intensitiesare related to the load cycle as illustrated in Figure 6.2. The opening and closure stressintensities are normally reported to be of the same order but not necessarily equal. Anexample can be found in the work of Packiaraj et al. [89], who investigated threshold valuesfor austenitic stainless steel of the type AISI 316, and as a part of this study they determinedexperimentally the opening and closure stress intensities. The obtained values of Kcl andKopen were observed to be fairly constant over the entire stress range investigated with 2.95� 0.43 and 4.14 � 0.55 MPa

pm being reported for Kcl and Kopen respectively. Also crack

length and R-ratio dependencies were found to be small. Chen [26] worked on crack closurein 0.13 % carbon steel. He tried to use both Kcl and Kopen in calculations of the e�ectivestress intensity range and both approaches were found to give a good correlating with theexperimentally obtained data.

6.2.1 Estimation of the Crack Opening Ratio U

The crack growth rate has been experimentally determined in terms of the e�ective stressintensity factor range for a large number of materials (see e.g. [132] where da/dN ��Ke�

relations are given for six common steel grades). In order to apply these data, it can beuseful to express �Ke� as a function of global loading parameters such as R or U , where thecrack opening ratio U is de�ned as

U =Kmax �Kopen

Kmax �Kmin=�Ke�

�K0:0 � U � 1:0 (6.2)


Figure 6.3: \Master diagram" for illustration of functional relationships between crack clo-sure levels and fatigue parameters.

McClung [82] carried out a critical review of experimental data and theoretical modelsfound in the literature for the description of a relationship between U and basic fatigueparametersK, applied stress �, and crack length a. Some of the observations and conclusionsmade by McClung will be outlined here. The opening behaviour was divided into the classes:I) near threshold, II) stable behaviour, and III) loss of constraint. In the near-thresholdregion, the opening level Kopen was generally reported to increase with a decreasing stressintensity factor level. Several mechanisms of closure may be operative in this region includingplasticity, surface roughness and oxidation. Above the near-threshold region in class II, theopening stress intensity Kopen becomes independent of the stress intensity factor level, butit is still a�ected by the applied stress level. The dominant mechanism is residual plasticity.Class III closure mechanisms are activated when extensive yielding is present around thecrack tip or in the remaining ligament. The elastic constraint is thus lost and the openingstress intensity drops to zero. To get an overview, McClung [82] suggests the construction ofa \master diagram" which shows the functional relationship between Kopen=Kmax and Kmax

or �K for a �xed load-ratioR and an applied load range. A sketch of such a diagram is givenin Figure 6.3. The \master diagram" is unfortunately encumbered with some disadvantagessuch as: 1) It does not demonstrate the interactions of applied stress and crack length withthe in uence of the stress intensity factors and 2) it does not show the R-dependency. Itshould be noted that there is a loose correspondence between the classes in the closuremap and the three regions of fatigue crack growth. However, a one-to-one correspondenceis not always observed. The overall conclusion from the literature review is that no singlerelationship between crack opening and fundamental fatigue parameters holds under allconditions, due to the wide range of parameters a�ecting the crack opening/closure.

Nevertheless, a practical estimation of U was derived by Tanaka and Soya [131] on theassumption of small-scale yielding. The e�ect of the plastic zone at the crack tip was takeninto account by shifting the actual location of the crack tip by half (rp) of the yieldingzone width ! as illustrated in Figure 6.4. For this con�guration, the linear elastic surface


Figure 6.4: Small-scale yielding model for closure analysis (sketch of Fig. 9 in [131]).

displacement v in the y-direction can by found by

v =4K�

E 0

srp � x

2�(6.3)

where K� is the stress intensity for the shifted crack position. Eq. (6.3) does not include thereduction of the v displacement arising from the plastic deformation of the crack wake. Aresidual plastic wake deformation �res was therefore proposed by Tanaka and Soya [131] andthe displacement along the surface thus becomes

vf = v � �res (6.4)

The crack opening stress intensity can be found by inserting Eq. (6.3) in Eq. (6.4) and thendemanding vf to be zero for x = 0:

Kopen =�resE

0

4

s2�

rp(6.5)

The e�ect of roughness and oxide-induced closure can also be included in the reduction ofthe opening displacement and �res then consists of three terms:

�res = �plastic + �oxide + �roughness (6.6)

where �plastic is known to be proportional to the opening displacement at the maximumload for R equal to 0, while it is less clear how �oxide and �roughness depend on the loadingparameters. Tanaka and Soya suggested that �plastic is calculated as a function f of Rmultiplied by the stress intensity range �K, and the sum of �oxide and �roughness is to be


taken as the displacement corresponding to the plastic deformation present when K reachesa constant value K0. These assumptions and Eq. (6.5) lead to

�res =4

E 0(f(R)�K +K0)| {z }

Kopen

rrp2�

(6.7)

where material properties are included in f(R) and K0. From Eq. (6.7) and Eq. (6.2) it ispossible to obtain the following expression for U :

U =1

1�R� f(R)� K0

�K= g(R)� K0

�K(6.8)

in which the function g(R) describes the R-dependency and it was assumed to take the form:

g(R) = 1=(R0 �R) (6.9)

where R0 is a constant representing material parameters such as the tensile strength. Theresulting expression for U can be written as

U =

(1:0=(R0 � R)� K0

�Kfor Kmin � Kopen

1:0 for Kmin > Kopen(6.10)

from which it is seen that U is approximated by a function of R and �K and two independentmaterial parameters K0 and R0. In another work Tanaka and Soya [132] presented K0 andR0 for six di�erent steel grades and these values are listed in Table 6.1.

Table 6.1: The material parameters K0 and R for six di�erent steel grades (from [132]).

Material �yield K0 R0

[MPa] MPapm

Steel-A 163.0 8.84 0.862Steel-B 269.0 9.66 0.886Steel-C 396.0 5.54 1.020Steel-D 578.0 3.99 1.030Steel-E 762.0 6.06 0.915Steel-F 888.0 4.15 1.030

The steel grades A to C and F were tested for stress ratios R in the range from 0.1 to0.8, while the steel grades D and E were tested in a larger range which also covered negative


R-values, going from -1.0 to 0.8. Eq. (6.10) gave a good correlation for all the materialstested [132]. By applying the material data from Table 6.1 and Eq. (6.10), Tanaka and Soyaobtained a single dA/dN -Ke� crack growth curve representing the six di�erent steel grades.The curve based on regression takes the following form:

dA=dN = C(�Kme� ��Km

e�;th) (6.11)

with C equal to 1:54 � 10�11, m = 2.75 and the e�ective threshold value determined as3.45 MPa

pm. Sumi et al. [125] simulated crack growth in the presence of biaxial loading

and residual welding stresses. They used a simpli�ed formulation of Eq. (6.10) where U isassumed to be a function of R only:

U =

(1:0=(1:5� R) for 0:0 < R � 0:51:0 for 0:5 > R

(6.12)

The crack was assumed to propagate in mode I condition and the corresponding crack growthdata for KA 36 steel was represented by:

dA=dN = 1:8 � 10�11h(U�KI)

m ��KmI;th0

i(6.13)

where �KI;th0 is the threshold stress intensity at R = 0 taken to be 2.45 MPapm and m is

equal to 2.932. No experimental observations were presented regarding the opening stressintensity levels. But a fairly good agreement was found in a comparison of the measuredand the simulated fatigue lives.

6.2.2 E�ective Stress Intensity Range under Mixed Mode Condi-tions

Several approaches have been proposed for the estimation of the e�ective stress intensityrange in mode I condition taking into account crack closure (as the two models outlined inthe previous section). Models for the estimation of crack closure in mixed mode conditions,however, are not very well described in the literature. Despite the fact that the closure e�ectis even more pronounced in mixed mode conditions than in pure mode I [142]. This probablyhas to do with the complex nature of mixed mode crack closure which, in addition to theparameters a�ecting mode I closure, is also in uenced by friction between the crack surfacesincluding wear.

Tong et al. [140] investigated the mean stress in uence on mixed mode threshold values.Mixed mode crack con�gurations were established by use of four-point bend specimens made


of structural steel BS4360 50D, and the obtained threshold values were compared with a the-oretical threshold boundary estimated by an equivalent stress range based on the maximumtangent stress criterion (see Eq. (4.27)). A fairly good agreement between experimental dataand the theoretical boundary was observed for R = 0.7, but deviations started to appear asthe stress ratio decreased and the e�ect of crack closure became more pronounced.

Crack propagation angles were determined as a part of the study. Here an interestingobservation was made by Tong et al. [140]. The crack propagation angles were found tobe insensitive to the mean stress level, which suggests that the closure a�ects equally bothmode I and mode II components of the stress intensity, leading to the following assumption[140]:

�KI,e�

�KI

=�KII,e�

�KII

(6.14)

Better correlation was obtained between the theoretical boundary and the experimental datawhen the e�ective stress ranges given by Eq. (6.14) were used. An experimental veri�cationof Eq. (6.14) by direct measurement of mixed opening stress intensities was unfortunatelynot carried out. Support of the model can, however, be found by the work of Baloch andBrown [12], who examined crack closure of near-threshold fatigue crack growth under mixedmode loading. They found that the e�ective parts of the mode I and the mode II stressintensity ranges were proportional to a mixed mode opening ratio U :

U = 1� dacd A

��KII

�KII + F�KI

�(6.15)

where dac is the amount of crystallographic crack surface, d is the average grain size of thematerial and A and F are arbitrary constants. These parameters have to be determined byexperiments for the given material, which makes the model unsuitable for the present study.

It has been argued that a larger part of the mode II range (the whole range [22]) shouldbe regarded as e�ective depending on the friction between the crack surfaces. In the presentinvestigations both the range described by Eq. (6.14) and the full range will be applied tosee how the mode II range a�ects the crack propagation in the case of crack closure.

6.3 Crack Closure by the Finite Element Approach

The closure of two crack surfaces can in a �nite element formulation be regarded as a contactproblem which is unfortunately non-linear by nature. A common way of dealing with contactproblems is to apply an impenetrability condition, i.e. a surface of one body is not allowed topenetrate the surface of another body. Two �nite element approaches can be used to satisfy

6.3. Crack Closure by the Finite Element Approach 93

the impenetrability condition: 1) Introduction of displacement constraints on the surfaceof the crack. This can be employed as a direct modi�cation of the sti�ness equations oras introduction of appropriate contact forces on the crack boundary. Before applying thedisplacement constraints, the points of contact must be identi�ed. The most straightforwardmethod is to specify a priori the node pairs which might come into contact during the loadcycle and then check the condition of these nodes [29]. 2) Use of special non-linear gapelements to couple the possible areas of contact. The gap elements change sti�ness with therelative position of the nodes which they couple. If the gap is open the sti�ness is set to zerowhile it is set to a large value if the crack is closed. Gap elements have with success beenapplied to linear elastic analysis of cracks [11, 16], they do not introduce new unknowns ordemand new solution strategies. The following discussion therefore concentrates on the useof gap elements.

6.3.1 Gap Element Formulation

The �rst step when gap elements are used is to identify the nodes which are to be coupled.This is relatively easily done in the case of a propagating crack. It is just to ensure an equalnumber of elements on the upper and the lower crack faces and then match the nodes ofthe two boundaries. In simulations of crack propagation, the gap elements are introduced atthe initial stage of the calculations and new elements are continuously added as the crack isextended during propagation. The contact state of each element has to be monitored duringthe load cycles. The condition (sti�ness) of the gap elements can be determined by thenormal strain � in each gap element (see Figure 6.5 for de�nition of the normal direction):

� > �1:0 : no contact, free state� � �1:0 : contact state (6.16)

Contact is obtained for � = �1:0 while � < �1:0 indicates \overlapping" between the cracksurfaces. The latter is not allowed and the system has to be recalculated with changed gapelement sti�nesses at the points where the surfaces are found to \overlap". Sliding mayoccur during the contact state if the friction between the crack surfaces is overcome. Thesimple Coulombs law is often applied with relatively good results for engineering problems[11]. Sliding takes place if � � �j�j, while the crack faces are assumed to stick together when� < �j�j, where � is the friction coe�cient and � and � are the shear and normal stressesin the gap element.

Properties of the Gap Elements

Many di�erent gap element formulations can be found in the literature. Bai and Zhao [8] havedeveloped gap elements for elasto-plastic calculations with large deformation using standard


Figure 6.5: Crack closure modelled with gap elements.

2-D and 3-D solid element formulations with modi�ed constitutive relations. Ballarini et al.[11] worked on a special element for the dilate frictional behaviour of rock discontinuities.The derived six-noded gap elements were successfully applied to both elastic-plastic andlinear elastic problems.

This discussion focuses on simple gap elements based on modi�ed truss elements neglect-ing the e�ect of friction (this is very similar to using springs between the crack surface asoriginally suggested by Newman [86]). If truss elements are to couple surfaces modelled byquadratic elements, it is important that the sti�ness of the gap elements between the \mid-side" nodes is twice the sti�ness of the corner nodes due to the shape function associatedwith the quadratic elements.

The sti�ness state of the elements is determined by Eq. (6.16) and in non-contact regionsthe gap elements should not o�er any resistance to the relative motion between the couplednodes. In contact areas, on the other hand, the sti�ness of the gap elements should besu�ciently large to prevent \overlapping" of the crack surfaces. An appropriate choice of gapelement sti�nesses will result in modelling of the two states and still avoid numerical trouble(extreme element sti�ness results in an ill-conditioned �nite element system). Newman [86]used springs with sti�nesses assigned to 107 times the elasticity modulus of the surroundingmaterial ES for the modelling of crack closure. A sensitivity analysis carried out in anotherstudy of plastic induced crack closure by Dougherty et al. [36] revealed that the model wasrelatively insensitive to the spring sti�ness. They applied springs with sti�nesses equal to1000 times ES. In the formulation by Bai et al. [8] the elasticity modulus E of the gapelements was assumed to be in the range 10�6 ES to 10�4 ES in the case of the free state.These values can be used as guidelines for choosing the gap element properties for a givensystem.


Convergence Criterion

The contact problem is non-linear and it therefore has to be solved by an iteration procedure.Convergence is considered to be reached when � � �1:0 for all the gap elements in the non-contact region. Convergence in the contact area is more complicated. Here a small overlapdescribed by � is accepted:

�1:0 + � � � � �1:0 (6.17)

where � is taken to be -0.02 as suggested in [8]. There is a large di�erence in gap elementsti�ness between the two states and for complicated contact problems an oscillation of gapelement states might occur. A maximum number of iterations is therefore employed in thepresent formulation to avoid a \never-ending" iteration procedure. An elegant, but morecomplex solution to the problem can be found in [29] where the element sti�nesses arecontinuously adjusted in the two states (for details see [29]).

6.3.2 Numerical Modelling of Geometrical Closure

Compressive residual stresses are known to close crack surfaces. An example of a cracklocated in a residual stress �eld is therefore used to test the capabilities of the gap elementformulation. A detailed description of the implementation of residual stresses in the numer-ical procedure will be presented in the subsequent chapter. This section focuses on some ofthe aspects regarding the application of gap elements.

Figure 6.6: Crack closure modelled with gap elements.

Beghini and Bertini [16] experimentally determined residual stress distributions for com-pact tension (CP) specimens cut from slabs with welding-induced stresses. The dimensionsof the specimens were manufactured according to the ASTM standards outlined in Figure6.6. A residual stress distribution obtained for a specimen with the thickness 12 mm and


Figure 6.7: Reproduction of residual stresses measured by Beghini and Bertini [16].

the parameter W equal to 48 mm is used in the present analysis. A reproduction of themeasured stresses acting perpendicularly to the crack direction is depicted in Figure 6.7 fora and y equal to zero. Compressive residual stresses can be observed in the two ends ofthe distribution while tensile residual stresses are found in the middle of the specimen. Twocon�gurations were analysed: 1) The crack tip located in the zone where the residual stresseschange from a compressive to a tensile state, and 2) the crack tip placed in the tensile regionof the residual stress �eld with a being equal to 6 and 12 mm respectively.

CP specimens are normally loaded through the holes by two pins by means of a tensilemachine, which in the numerical analysis was approximated by two concentrated nodal pointloads. The loads P (assumed to be tensile) open the crack while the compressive residualstresses try to close the crack faces. An equilibrium for a given length of crack closure has tobe obtained by iteration with gap elements employed to prevent \overlapping" of the crackfaces.

Figure 6.8: Di�erent stages of crack closure for the CP specimen with a equal to 6 mm.


Figure 6.8 shows three stages of closure for an increasing load level P acting on the \shortcrack" con�guration: 1) Complete closure, 2) partial closure of the crack faces with the crackbeing open in the region of the crack tip, and 3) a completely open crack. The lines betweenthe crack faces in the open stages indicate the gap elements.

The relation between the intensity due to the externally applied loads P and the e�ectivestress intensity is depicted in Figure 6.9 for a equal to 6 mm and in Figure 6.10 for a cracklength equal to 12 mm. The di�erent slopes of the curves indicate the stages of crack closure.The crack opening stress intensity level can be estimated from the slope of the curves. Itis located at the bend after which the e�ective stress intensity and the externally appliedstress intensity become proportional. The crack opening level Kopen is rather blurred for theshort-crack con�guration but estimated to be equal to 12 MPam1=2. For a located in thetensile zone it is relatively easy to detect an approximate value of Kopen which is also equalto 12 MPam1=2. Beghini and Bertini [16] both experimentally and numerically determinedKopen for a whole range of crack lengths, and they found that the opening stress intensityvaries little with the crack length as observed in the present analysis.

Figure 6.9: The externally applied stress in-tensity versus the e�ective stress intensity fora equal to 6 mm.

Figure 6.10: The externally applied stress in-tensity versus the e�ective stress intensity fora equal to 12 mm.

Some numerical problems occurred during the analysis. Convergence could not be ob-tained for some of the partly closed con�gurations. The sti�ness of a few gap elementscontinued to shift between the two states during the iterations. This problem could to somedegree be solved by changing the sti�ness of the gap elements, but the sti�ness had to bekept within some limits, e.g. the sti�ness of an open gap element should not a�ect thedetermination of the stress intensity at the crack tip. So convergence was not obtained forall con�gurations of partly closed cracks, which explains the location of the marked datapoint in Figure 6.9.


6.3.3 Comments on Element Modelling of Crack Closure

The �nite element method can be applied to an idealised closure problem as it is seen from thejust described example of geometrical closure. Plastic-induced closure can also be modelledby use of the �nite element method (see e.g. [36, 83, 81]) and qualitative understanding ofclosure mechanisms can be obtained by these models. But �nite element modelling of allthe factors a�ecting closure is rather di�cult. Therefore, the empirical relations outlined inSection 6.2.1 will be applied to modelling of crack closure.

6.4 Summary

Crack closure is a�ected by a large number of factors such as plasticity, oxide on the cracksurfaces, and the roughness of the crack surfaces. A qualitative understanding of di�erentclosure mechanisms can be obtained by the application of �nite element models. However,establishing a model which can account for all parameters which in uence crack closure israther di�cult. Hence, in the present formulation, crack closure is modelled by empiricalformulas relating the e�ective stress intensity range to the stress intensity at the crack tip.

Empirical models for determination of the e�ective stress intensity range have been out-lined. However, there is some confusion as regards the determination of the e�ective modeII stress intensity range. Two models are applied: 1) The e�ective range which follows thee�ective mode I range and, 2) the whole mode II range is regarded as e�ective. The twomodels are assumed to represent the boundaries of the e�ective mode II range.

Chapter 7

E�ect of Residual Stresses on Crack

Propagation

7.1 Introduction

Residual stresses are self-equilibrating stresses existing in materials or structures under uni-form temperature conditions. This type of stresses is present in most engineering componentsas a product of assembling or manufacturing procedures such as ame-cutting, welding andcold forming. These processes cause residual stresses by inducing plastic deformation of themetal through severe temperature gradients, mechanical forces or microstructure changes[50].

One of the major interests in residual stresses is their e�ect on the fatigue life of structuressubjected to cyclic loading. It is well known that compressive stresses retard the fatiguegrowth and in some cases even cause crack arrest, while the opposite e�ect can be observedin regions of tensile stresses. The ability of retarding the fatigue growth by compressivestresses is used to extend the fatigue life of components by techniques such as shot-peening,which is high-velocity bombarding of a surface with shots of steel, glass or ceramics inducingcompressive residual stresses near the surfaces of the components. Other surface treatmentswhich create compressive residual stress are described in [44].

Establishing a representative initial stress distribution is an important part of the analysisof the e�ect of residual stresses and the next section is therefore devoted to the determinationof residual stress �elds. Given a residual stress �eld, it is possible to analyse the in uenceof the residual stresses on a propagating crack. The residual stress e�ect can be accountedfor by a modi�cation of the stress intensity factors by use of the superposition method asdescribed in Section 7.3.1. Special attention also has to be paid in evaluations of the crackgrowth rates for cracks propagating in residual stress �elds, which will be discussed in Section7.5.

99

100 Chapter 7. E�ect of Residual Stresses on Crack Propagation

Investigation of mixed mode fatigue crack propagation in residual stress �elds is a com-plicated subject with many still unsolved problems. To achieve some practical results, thepresent formulation is therefore restricted by the following assumptions:

� The initial residual stress distribution is only a�ected by the propagating crack, thusno redistribution of the initial �eld is introduced due to shakedown or overloads.

� Only residual stresses arising from welding and ame-cutting are dealt with.

� The external loads are restricted to constant amplitude cyclic loading.

� E�ects of multipass welding is not taken into account.

7.2 Residual Stress Distributions

The �rst step in analysing the e�ect of residual stresses is to establish a residual stress�eld. The magnitude and the distribution of the residual stresses can be found by di�erentexperimental measuring techniques. These techniques can be divided into destructive andnon-destructive methods. Two attractive non-destructive approaches are : 1) X-ray stressanalysis, primarily used for the determination of surface stresses since the applicable X-raywave lengths have only very short penetration depths (typically 5-10 �m in most materials[74]). The method has the advantage that it can be applied to very small surface elements,and it can therefore be used for determination of residual stresses in notches and areas closeto the crack tip [76]. 2) Neutron di�raction method, which can be used for determination ofinternal stresses. Neutrons are scattered by the atomic nuclei which results in a far deeperpenetration than X-rays scattered by electron shells. For many structural materials it is thuspossible to investigate internal residual stresses in regions up to several centimetres insidethe components (3 cm in steel and 30 cm in aluminium alloys [97]). The more traditionaldestructive mechanical approaches, like the releasing slice and the hole drilling techniques,have the obvious disadvantage that they destroy the structures to be analysed. Nevertheless,these methods are widely used due to their accuracy and relatively low cost.

The �nite element method o�ers another suitable tool for the evaluation of residual stress�elds and it can be applied together with experimental measurement. Beghini and Bertini[17] developed a technique for modelling of plane residual stress �elds. Relatively few straingauge measurements were combined with �nite element calculations. The method was testedon several specimens with di�erent residual stress con�gurations and good descriptions ofthe stress �eld were obtained [17].

Ueda and Fukuda [143] worked on a model for three-dimensional determination of theresidual stress �elds. The distribution of inherent strain (by de�nition the strain whichcauses residual stresses) does not change unless new plastic strains are introduced. Basedon this observation, Ueda and Fukuda suggested establishing two-dimensional strain �elds

7.2. Residual Stress Distributions 101

from pieces cut out of the welded structure. By use of inherent strain as a parameter theytransformed two-dimensional strain measurements into three-dimensional strain �elds. The�nal residual stress distribution was obtained by using the inherent strain �eld as initialstrains in a linear elastic �nite element calculation.

Finite element analysis can also be applied to simulation of a complete manufacturingprocess (such as a welding procedure) using elastic-plastic calculation, temperature elementsand information about the temperature distribution in the material during the process [108,109, 144, 145, 146]. But all the methods listed above are either based on measurementsor "time-consuming" elastic-plastic �nite element calculations. For structures composed ofthin butt- or �llet-welded plates and sti�eners, simpli�ed approaches have proved to predictresidual stresses with reasonable accuracy. One of these approaches is a method proposed byYuan and Ueda [152], who have shown that welding-induced residual stresses in structuralmembers like I- and T-pro�les can be estimated by a combination of simple formulas forinherent strain and elastic �nite element calculations. The components of this approach areoutlined in Section 7.2.2 while an estimation of residual stresses arising from ame-cuttingis discussed in Section 7.2.3.

7.2.1 Welding-Induced Residual Stress Fields

Welding generates a highly localised transient heat input which causes the weld area to beheated up sharply relative to the surrounding material. A main source of residual stressesis produced by the following shrinkage of the weld seam, which exhibits resistance to de-formation at the cooling stage from the adjacent material. The second important sourceis related to volume expansion arising from phase changes in the weld. These two sourcesinduce residual stresses in the direction of the weld, and these stresses are here called lon-gitudinal stresses while the stresses generated in the direction perpendicular to the weldare called transverse stresses. The present formulation is restricted to thin plates, wherethe temperature distribution is assumed to be uniform through the thickness and where thestress variation through the thickness of the plates is neglected.

One of the simplest cases of welding-induced residual stresses is the joining of two mildsteel plates by a central weld. An idealisation of this case is here used to illustrate how resid-ual stresses can be formed by shrinkage. Figure 7.1 shows a sketch of the residual \shrinkagestresses" arising from the joining of the plates. Typical for this kind of weld is the longi-tudinal tensile zone located in the narrow area close to the weld. The maximum value ofthe tensile stress zone is normally at or slightly above the yield limit. Lower compressivestresses are found in the adjacent region, dropping o� rapidly as the distance to the weldincreases. Transverse stresses are produced by the transverse contraction of the weld seamand indirectly by the longitudinal contraction of the seam. Along the weld, tensile stressesof relatively low magnitude (normally �22 < 0:2�11 [149]) are produced in the middle partand compressive stresses in the ends. This transverse stress distribution can be changed


Figure 7.1: Schematic presentation of longitudinal and transverse stresses in a rectangularplate with a central butt weld.

signi�cantly by displacement constraints at the edges of a plate while the longitudinal com-ponent is less sensitive to boundary conditions [80]. The symmetry of the distribution in thelongitudinal direction applies only if the welding speed is very high compared to the rate ofheat conduction of the material.

The second source of residual stress is related to microstructural changes in the material.A phase transformation (for instance a ! �-transformation) may take place during thecooling period and cause an expansion of the material. If the yield stress is su�ciently highduring the time of phase change, compressive residual stresses can be generated in the weldarea. The �nal distribution is the superposition of the two types of stress �elds. Longitudinal

Figure 7.2: Sketch of longitudinal stress distribution for three di�erent welds.

stress distributions are sketched in Figure 7.2 for three di�erent types of weldings. Case a)is for for normal mild and low-alloy steel, where the "shrinkage" stress is the dominantcomponent, which results in a tensile zone in the middle with a magnitude close to the yieldlimit. Compressive stresses due to the transformation process can be found in aluminium


Figure 7.3: The two analytical estimates of the residual stress distribution and a reproductionof the experimental results by Kanazawa et al. (from [135]).

alloys and high-alloy steels. The peak value in the tensile region is thus reduced as sketchedin case b). The stress distribution given in case c) is for a high-alloy steel with ferritic weldmaterial where signi�cant phase change has the result that the stresses in the weld centre gointo the compressive range [97]. This formulation concentrates on welding residual stressesappearing in mild and low-alloy steels, i.e. case a).

Longitudinal Residual Stress Distribution for a Butt Weld

When the magnitude of the residual stresses arising from a butt weld is considered, it isseen that the longitudinal stress is the main component. Investigations of the transversedistribution of longitudinal stresses have been carried out by several authors. Terada [134]was one of the �rst to establish a theoretical estimation of the longitudinal residual stressesarising from the joining procedure. He stated the following requirements for the stress �eldf : (a) The stress must result in zero axial force, (b) f(�) has to be an even function becauseof symmetry, where � is the coordinate perpendicular to the welding line normalised by thehalf length of the tensile region, (c) maximum stress �0 is to appear at the weld, (d) thee�ect of the weld has to vanish far from the weld (f(�) = 0 for � = �1), and (e) f(�) is todecrease monotonically from the centre of the weld until it reaches some negative minimumvalue and then f(�) is to increase monotonically until zero is reached again. To meet theserequirements, Terada chose the following function:

f1(�) = �0e�0:5�2

�1� �2

�(7.1)

Figure 7.3 shows a reproduction of experiments performed by Kanazawa et al. (from [135])compared to the residual stress distribution described by Eq. (7.1). A good agreement isfound in the tensile area, but too large stresses are estimated in the compressive region.


Tada and Paris [127] suggested another function for the estimation of the residual stress�eld due to a butt weld, also based on the requirements listed by Terada:

f2(�) = �0(1� �2)

(1 + �4)(7.2)

This solution gives a good description of the tensile region (see Figure 7.3), whereas Eq. (7.2)predicts too low stresses in the compressive region. Despite the problems in the compressiveregion, both f1 and f2 give reasonable estimations of the residual stresses measured byKanazawa, and the two analytical distributions will be used in test examples later on.

7.2.2 Simpli�ed Prediction of Residual Stresses by Use of InherentStrain

Inherent Strain Distributions

Carrying out experiments or complete thermal elasto-plastic calculations is very expensiveand time-consuming for complex structures. Approximation procedures therefore often haveto be applied to get some practical results. A simpli�ed procedure for a qualitative estimationof longitudinal residual stresses in thin wall structures is outlined here. The method wasformulated by Ueda and Yuan [146] and is based on inherent strains. These strains canbe used as an input to a linear elastic determination of the residual stress distribution in astructure. Ueda and Yuan carried out some numerical experiments in order to study inherentstrain distributions in butt welds. Figure 7.4 shows a schematic representation of a computed

Figure 7.4: Schematic representation of inherent strains in a single butt welded plate (from[146]).


inherent strain distribution along a butt weld. It is seen that the shape of the longitudinalstrain distribution (��x) is prismatic in the middle part of the weld where the e�ects of thefree ends do not interfere. Ueda and Yuan found that the shape of the inherent strain fora cross-section located in the middle part of the weld may be approximated by a trapezoid.Their work showed that the inherent strain pattern varies little with the welding conditionsand the size of the plate. The observations made formed the basis for the development ofsimple formulas for the description of the longitudinal inherent strain, taking into accountwelding conditions, geometrical dimensions and material properties. The procedure wasvalidated by thermal elasto-plastic calculations [152].

The formulation focuses on the distribution of longitudinal inherent strain. From thesketch given in Figure 7.4 it is noted that the transverse inherent strain ��y only exists in theregions a�ected by the free ends. The same holds for shear strain (not in the sketch), whichalso only appears in the vicinity of the ends. The inherent strain distribution illustratedin Figure 7.4 is for a given set of boundary conditions, and di�erent boundary conditionscould change the strain distribution. However, the longitudinal ��x is the dominant straincomponent in thin-plated structures and it is believed to cover the major e�ect of residualstresses. Therefore, disregard of the e�ect of ��y and �

�

xy is accepted in the present formulation.The biggest error from this assumption will usually occur in the ends of the welds.

By further numerical studies, Yuan and Ueda extended their method for determinationof inherent strains and residual stresses in butt welds to more complicated details such asT- and I-pro�les [152]. It was found that the inherent strains introduced by the weld in the ange side and in the web side of the �llet welds are almost the same, provided the widthsof the plates are not very di�erent. The simple formulas for prediction of inherent strains inbutt welds were modi�ed for estimation of inherent strains in the T- and I-pro�les, takinginto account new temperature distributions and bending deformations. Predicted residualstress distributions were compared with experimentally obtained distributions and a goodagreement was found.

Estimation of Longitudinal Residual Stresses in a Butt Weld by Use of InherentStrain

The longitudinal inherent strain distribution in the transverse section of a butt weld can beapproximated by a trapezoid as illustrated in Figure 7.5a. On the basis of the numericalstudies, Ueda and Yuan [146] established some formulas for the description of the charac-teristic values of the approximated distribution, i.e. the peak value �̂�x, the width b, and thewidth of the heat-a�ected zone (HAZ) yH . The following formulas were derived for a buttjoint in a plate with a width of 2B, assuming that the heat source is applied instantaneouslyto the entire weld line (y = 0), so that the material deforms uniformly perpendicularly to


the weld. The formulas have the following form [152]:

yH = 0:242 Q=[c�h(Tm � T0)]

b = � b0

�̂ �x = ��̂�x0

(7.3)

and

b0 =�EQp

2e� c�h�YB=0:242�EQ

c�h�YB

�̂�x0 = �YW=E

� = 1� 0:27�ETav=�YB

� = � � 2 = �1� 0:27�ETav=�YB

Tav = Q=c�A

(7.4)

where

Q = line heat input (J/m)Tm = mechanical melting point over which yield stresses disappear (oC)T0 = room temperature (oC)c = speci�c heat (J/kgoC)� = density (kg/m3)h = thickness of the plate (m)� = linear thermal expansion coe�cient (1=oC)E = Youngs modulus (Pa)�YW = yield stress of weld metal and HAZ (Pa)�YB = yield stress of base material (Pa)A = area of the transverse cross-section (m2)

The predicted strain �eld can be used as input to the calculation of the longitudinal residualstress distribution in the welded structure. A simple example of a resulting stress distributionis shown in Figure 7.5b, where it is assumed that the residual stress at the weld is equalto the yield stress of the weld metal, �YW . The magnitude of ��x is found from the forceequilibrium condition (

R B0 �xhdy = 0).


Figure 7.5: Simpli�ed inherent strain and residual stress distributions.

Inherent Strain Method Applied to T- and I-Joints

Further numerical studies carried out by Yuan and Ueda have shown that the formulasoutlined in the previous section can be applied to I- and T-joints if some small modi�cationsare made [152]. The magnitude of the inherent strain is practically the same for a butt weldand a T-joint, and it is therefore only the width of the inherent strain zone controlled by b0and � which has to be modi�ed. The heat input in a T-joint is normally di�erent from theheat input in a butt joint. This e�ect can be accounted for by replacing the heat inputQ usedfor the determination of b0 in Eq. (7.4) by an equivalent heat input Q0 = 2Qh=(2 hf + hw)(see Figure 7.6) which results in a new estimation of b0:

b0 = 0:484�EQ=[c�(2hf + hw)�YB] (7.5)

where hf and hw are the plate thicknesses of the web and the ange respectively.

Figure 7.6: Sketch for determination of equivalent heat input.

In the case where the average temperature rise is larger than 300C, signi�cant bendingdeformations may occur as a consequence of the non-uniform heat input in the web of the T-or I-pro�le and a second modi�cation has to be made. It starts by dividing the longitudinalstrain in the ange into two terms: a) Uniform deformation and b) bending deformation.Inherent strains may be in uenced by both types of deformations, where the e�ect of theformer can be evaluated by the formulas derived for butt welds. Yuan and Ueda [152] foundthat the moment introduced by the non-uniform temperature distribution in the web isproportional to the average temperature in the whole section:

M � �ETavA�z (7.6)


where �z is the distance between the weld centre and the neutral axis of the transversesection. The corresponding longitudinal strain becomes �0x =

MEI�z ' �Tav

A�z2

Iwith I being

the moment of inertia for the cross section. Therefore, the bending part can be accountedfor by increasing the average temperature in the determination of �:

T 0av = Tav +�Tav � (1 + �)Tav

� = �z2A=I(7.7)

so that

� = 1� 0:27 �ETAV (1 + �)=�Y B (7.8)

An estimation of the longitudinal inherent strain in a T-pro�le can now be found byapplying Eq. (7.3) and Eq. (7.4) to the determination of the magnitude of the strain, whileEq. (7.7) and Eq. (7.8) are used to �nd the distribution.

The inherent strain distribution can also be found in an I-pro�le by means of the pro-cedure described above. If the welding sequence is to be included in the determinationof the residual stresses, the following procedure is suggested by Yuan and Ueda [152]: a)Inherent strains are imposed on a stress-free T-pro�le so that �0 is obtained, b) the sameinherent strain distribution is applied to a stress-free I-pro�le, which gives �00, and c) thetwo stress �elds are superposed which results in the �nal longitudinal residual stress dis-tribution. Numerical experiments were carried out for a typical ship section (by use of themethod described above), they showed that the stress distributions in the upper and lowerparts of the pro�le are almost the same. This indicates that the two inherent strain �eldscould be applied simultaneously, neglecting the e�ect of welding sequence and thus reducingthe calculation time without any signi�cant errors in the determination of the residual stress�eld. It should, however, be noted that the welding sequence in the case of some problemsmay have a signi�cant in uence, especially on the welding-induced de ection.

Comments on the Inherent Strain Method

The methods for estimation of longitudinal inherent strain outlined in the previous sectionsare developed for thin plates. The de�nition of \thin plates" in this context is plates in whichthe temperature distribution and other quantities of interest can be regarded as uniformthrough the thickness during welding. The methods have been validated for plate thicknessesup to 12 mm but no upper limit has been given. Wu and Carlson [149] carried out a studyof stress intensity factors for a half elliptical surface crack located in thin and thick platescontaining residual stresses. They found that the thickness e�ect can be disregarded forplates with a thickness below approximately 25 mm.

Welding speeds are usually in the range of 0.001-0.03 m/s, which is relatively high com-pared to the rate of heat conduction [149]. This justi�es the assumption about the heatsource being applied instantaneously to the entire weld line.


7.2.3 Residual Stresses from Flame-Cutting

Gell investigated the redistribution of residual stresses in ame-cut and welded thin plates[47]. He found that the e�ect of ame-cutting with an oxygen-acetylene ame is similar tothat of welding. A tensile zone is created in the vicinity of the edge similar to the tensilezone along a welding line. Thus, if the heat input is known, the size of the yield zone maybe estimated in the same way as for the welding procedure.

However, an accurate prediction of the heat input is rather di�cult, especially for thinplates. Alternatively, the residual stresses arising from ame-cutting can be based on a\tension" block method where a block of shrinkage force is loaded into the model (by directinput of strains or by thermal loading). Gell [47] outlined and investigated several proposedmethods found in the literature for the determination of the width of the tensile block cin the case of ame-cutting. The formulas were based on experimental work relating theshrinkage force to the plate thickness t. One of the formulas has the following form:

c = 1100

pt

�Yield(7.9)

where c and t are in mm and the yield stress �Yield of the plate is in MPa. It should bementioned that these procedures are only valid for low-carbon steels, because, for high-carbon steels, compressive stresses can be created in a zone close to the surface (the widthof this zone is approximately 0.5 mm to 1.0 mm) due to martensitic hardening [97].

7.2.4 Redistribution of Residual Stresses

A redistribution of the initial residual stress �eld is often found in cyclically loaded structuressuch as ships and o�shore platforms. The redistribution can occur due to an extreme loadingof the structure, which causes additional plastic stresses to be induced, or the redistributioncan be a consequence of \shakedown". Shakedown is when the residual stresses graduallydisappear in cyclically loaded structures [23]. A large number of factors may a�ect theresidual stress distribution including: (a) The mechanical properties of the material, (b)the direction and the amplitude of the cyclic loading, (c) the number of load cycles, (d)the direction, the level and the gradient of the residual stresses, and (e) the temperature.Thus, a theoretical analysis of the redistribution of initial residual stresses is a complicatedmatter and the redistribution is therefore encumbered with a great number of uncertainties.Nevertheless, Eckerlid and Ulfvarson [37] carried out one-dimensional fatigue investigationsof cutouts in ship structures, in which they included redistribution of residual stresses duringthe lifetime of the ship. An interesting observation was that, after one voyage in ballastcondition followed by one voyage in fully loaded condition, the residual stress distributionbecame stable according to their calculations. Still a lot of uncertainties are to be solved.Therefore, due to the complexity of multiaxial loading, the present formulation is restrictedto a static elastic residual stress �eld only a�ected by the propagating crack.


Figure 7.7: Superposition technique for a cracked body containing residual stresses.

7.3 Stress Intensity Factors in Residual Stress Fields

The initial residual stresses are in general redistributed when a crack is introduced in a bodycontaining residual stresses. An example is a surface without external loads where a physicalrequirement is that the stresses normal to the surface are zero. To satisfy this condition fora crack which propagates into a residual stress �eld, the initial stress �eld normally has tobe redistributed. Thus, a model for crack path simulation in residual stress �elds has to takeinto account the redistribution of stresses caused by a propagating crack.

7.3.1 Superposition Method

The superposition technique is a common and relatively simple way of dealing with linearelastic residual stresses and their redistribution. This technique states that the elastic stress�eld due to two or more load systems can be regarded as the sum of the contribution fromeach stress �eld. Thus, the residual stress �eld is simply added to the externally inducedstress �eld. The redistribution of the residual stresses due to a propagating crack is accountedfor by decomposing the total stress �eld into appropriate part solutions, which satisfy theboundary conditions. It should be noted that the superposition principle is valid after theredistribution of stresses due to the introduction of a crack [92, 139].

Figure 7.7 shows the superposition technique applied to a cracked body which is subjectedto the external load T on the surface ST , a residual stress �eld given by �R and �R, and adisplacement boundary condition V on the surface SV . The system is decomposed into threeload cases: (1) An un-cracked body containing the residual stress �eld. (2) A cracked bodywith the crack faces being subjected to the reverse sign stresses. The reverse sign stressesare obtained at the line of the potential crack in the un-cracked body without applicationof external loads. (3) A cracked body subjected to external loads only. The stress intensityfactors (SIFs) for a crack with closed crack faces are equal to zero. The stress intensityfactor due to the residual stress �eld thus becomes equal to the contribution from the loaded

7.3. Stress Intensity Factors in Residual Stress Fields 111

crack faces (loaded with ��R and ��R). SIFs due to external loads are computed by normalprocedures, and the resulting stress intensity factor simply becomes: KR +KEx.

7.3.2 In uence Function/Weight Function Method

The theoretical stress intensity factor for a body subjected to residual stresses on its crackfaces can be computed by use of the in uence function method (also called the weightfunction method). The stress intensity factor due to an arbitrary stress distribution �R onthe crack surfaces is calculated from an integral:

KR =Z a

�a�Rm(x)dx (7.10)

where a� and a+ de�ne the two crack tips and the weight function m(x) represents the stressintensity factor for a pair of unit splitting forces P acting perpendicularly to the crack surface,as illustrated in Figure 7.8. The in uence function method is an exact method provided

Figure 7.8: Splitting forces P acting perpendicularly to the crack surface.

that the correct weight functions are used. These functions can be obtained analytically,by experiments or by use of the �nite element method. Glinka [50] has collected analyticalweight functions which are useful for common through thickness crack con�gurations. Thein uence function method makes it possible to calculate stress intensity factors for a varietyof crack-residual stress distribution con�gurations, but the complexity of the weight functions[50] often requires a numerical evaluation of Eq. (7.10). It should be mentioned that whenEq. (7.10) is used for compressive stress �elds, negative KI values may be obtained which arenot meaningful unless some additional external tensile loads are applied (negative KI valuesindicate non-physical overlapping of the crack surfaces).

7.3.3 Stress Intensity Factors from Residual Stress Fields by theFEM

The stress intensity factors due to residual stresses can also be determined by use of the�nite element method. Di�erent approaches are available: (1) A residual stress �eld which


Figure 7.9: Boundaries for the domain integral.

produces self-equilibrating forces and moments can be introduced as initial stresses/strains inthe elements in uenced by residual stresses, or (2) the superposition method can be appliedand the crack surfaces are subjected to the loads arising from the reverse sign stresses. Thesuperpositon method with the crack face loads was found to be easiest to implement in theexisting part of the crack simulation procedure and was therefore chosen.

Implementation of Crack Face Loads

The computation of SIFs by use of the Displacement Correlation Technique (DCT) is nota�ected by the introduction of a residual stress �eld as crack face loads. But the evaluationsof the J-domain integral de�ned by Eq. (5.11) are. Raju and Shivakumar [98] have shownthat the transformation of the J-contour integral into a domain integral (see Section 5.3.4)also results in line integrals along crack surfaces which are zero in the case of traction-freefaces. However, the line integrals may contribute signi�cantly to the J-integral when crackface loads are applied. Thus, the calculation of the J-integral must include both the domainand the additional line integrals, and the latter have to be evaluated along the crack facesfrom the outer boundary of the domain to the crack tip. When the decomposition methodis used, the line integrals have the following form [98]:

JI1;Line = 2Z C

F�22

@u2@x1

qdx1 + 2Z 0

C�22

@u2@x1

dx1 (7.11)

JII1;Line = 2Z C

F�12

@u2@x1

qdx1 + 2Z 0

C�12

@u2@x1

dx1 (7.12)

where �22 and �12 are stresses acting on the crack faces. The boundaries of the integrals aredepicted in Figure 7.9.

7.4. Input of Residual Strain and Stress Distributions 113

Figure 7.10: Input of inherent strains for determination of the residual stress �eld.

7.4 Input of Residual Strain and Stress Distributions

In the present formulation the residual stress �elds can be given as a direct input or they canbe calculated by the inherent strain method. The input is related to an edge in the model,which makes it easy to specify stress and strain distributions caused by ame-cutting orwelding of two edges. The distributions can assume abitrary shapes (as functions or discretedistributions). An example of a possible input for inherent strain calculations is given inFigure 7.10.

When the inherent strain approach is applied, the crack surfaces are \stitched" togetherand the residual stresses arising from the inherent strains are determined in an un-crackedbody which is the basis for the superposition method.

7.5 Crack Growth Rates

In the presence of residual stresses, the e�ective part of the stress range Ke� can di�ersigni�cantly from the externally exposed stress intensity range KEx. In general the di�erenceis not only a function of the residual stresses, but also of the applied load, since the crackpropagation could be a�ected by crack closure.

The \e�ective stress ratio", R0, originally proposed by Glinka [49], is a commonly usedmethod [16, 19, 67, 68, 125] to account for the residual stress e�ect in the mode I crackgrowth. The e�ective stress ratio is simply de�ned as [125]:

R0 =

8<:

KI,min+KI,RKI,max+KI,R

for KI,min +KI,R � 0

0 otherwise(7.13)


The corresponding e�ective mode I stress range can be determined by

�KI =

8><>:

KI,max �KI,min for KI,min +KI,R � 0KI,max +KI,R for KI,max +KI,R � 0 and KI,min +KI,R � 00 otherwise

(7.14)

An e�ective mode II stress ratio could be de�ned in a way similar to Eq. (7.13), but to theauthors knowledge, it is not yet clear how to de�ne an e�ective mixed mode stress ratio. Themode I stress intensity factor is known to dominate the fatigue crack growth rate. Therefore,it was decided to apply the e�ective stress ratio given by Eq. (7.13) to mixed mode crackpropagation. In mode II, stress intensity factors can without problems assume negativevalues and normal procedures can be applied in the determination of the mode II stressintensity range. An e�ective range could be determined by Eq. (6.14) as described in theprevious chapter.

7.5.1 Crack Growth in Transition from Compressive to TensileResidual Stress Fields

Kang et al. investigated mode I crack growth in both tensile [67] and compressive [68] residualstress �elds. They found that the e�ective stress ratio (the R0-method) can successfully beapplied to predict fatigue crack growth in tensile residual stress �elds even under compressivecyclic loading. The study of cracks growing in compressive residual stress �elds showedthat the R0-method can be used for estimation of crack growth in regions of almost uniformcompressive residual stresses. On the other hand, in regions with transition from compressiveto tensile residual stresses an error may occur in the application of the R0-method. This isrelated to partial closure of the crack faces [68].

However, the detailed behaviour of crack growth/closure of crack propagating throughtransition regions is not yet clear. Therefore, the e�ects of the transition are neglected inthe present study.

7.6 Test Examples

The theory outlined in the previous sections was implemented in the crack simulation pro-cedure. Some examples of calculated stress intensity factors and di�erent crack propagationscenarios will be given in the following. The �nal example is of a three-dimensional plateassembly where the e�ects of welding-induced residual stresses are included. Both crackpaths and estimated fatigue lives are compared with experimentally obtained results.

7.6. Test Examples 115

7.6.1 Crack Located Symmetrically with Respect to the WeldLine

The �rst example is stress intensity factors for a crack located symmetrically about a butt-weld line (L = 0 in Figure 7.11) with the longitudinal stresses due to the weld described bythe functions given in Section 7.2.1. Stress intensity factors were numerically determinedby means of both the displacement correlation technique and the J-integral method. Theobtained results are compared with the SIFs calculated by the in uence function method.In the case of a small crack located in a wide plate the weight function m(x) derived for a

Figure 7.11: Crack in a longitudinal residual stress �eld.

crack in an in�nite plate can be used [50]:

m(x)�a =1p�a

sa� �

a� �(7.15)

where �a refers to the two crack tips as illustrated in Figure 7.11. Both a and the parameterL denoting the eccentricity have been normalised by half the width of the tensile zone Lt.By insertion of the weight function given by Eq. (7.15) and a residual stress �eld describedby either f1 or f2 into Eq. (7.10), the following expression for the SIFs is obtained:

K�a =1p�a

Z a

�afi(�)

pa� �pa� �

d� i = 1; 2 (7.16)

Figure 7.12 and Figure 7.13 show the stress intensity factors normalised by �0p�a as

a function of the crack length computed for a crack which is located in the two residualstress �elds f1 and f2. The crack length is normalised by the length of the tensile zone ofthe residual stress distribution and it ranges from 0.5 to 1.0. This implies that the crack isalways located inside the tensile zone. The two �gures show that the results obtained by theDCT are in very good agreement with the SIFs computed by the in uence function method.Moreover, the J-domain integral solution provides a good estimation of the SIFs in most ofthe displayed crack length interval, but a small divergence (5 %) is observed for cracks ofsmall lengths.


Figure 7.12: SIF as a function of the cracklength due to the residual stress �eld de-scribed by Terada [134].

Figure 7.13: SIF as a function of the cracklength due to the residual stress �eld de-scribed by Tada and Paris [127].

Table 7.1: SIF due to residual stress �eld described by Eq. (7.1) and Eq. (7.2), a = 1:0; L =1:0.

f1 by Eq. (7.1) f2 by Eq. (7.2)Ka�=�0

p�a Ka�=�0

p�a

Theoretical (Eq. (7.16)) 0.5913 0.6269DCT 0.5885 0.6241J-integral (EDI) 0.5672 0.6028

7.6.2 Crack Located Eccentrically with Respect to the Weld Line

In the �rst example the crack tip was always located in the region of tensile residualstresses. In this example, the crack is given a displacement along the � axis, and one endof the crack is then located in the compressive zone, while the other crack tip is locatedin the tensile region. The calculations were carried out for a = 1:0 and L = 1:0 (seeFigure 7.11 for the notation). The computed stress intensity factors obtained by the threemethods are presented in Table 7.1 for the crack tip which is located in the tensile region.A good agreement is observed between the SIFs calculated by the three methods. But, fromthe table it can again be seen that the displacement correlation technique provides slightlybetter results than the J-domain integral method in a comparison to the results from thein uence function method.

7.6.3 Mixed Mode SIFs in Residual Stress Fields

The previous examples are restricted to cracks located in mode I residual stress �elds. But ingeneral the crack propagates through mixed mode residual stress �elds. No relevant practical


examples have been found in the literature for cracks located in mixed mode residual stress�elds. The con�guration illustrated in Figure 7.14 is therefore used to verify the numerical

Figure 7.14: Centre crack located in a mixed mode stress �eld.

implementation of the superposition method for the mixed condition. The theoretical stressintensity factors for a crack located in an in�nite biaxially loaded plate can be computedfrom:

KI =��22 sin

2 � + �11 cos2 ��p

�a

KII = (�22 � �11) sin� cos �p�a

(7.17)

In the numerical models, the width and the height of the plate are equal to 25 timesthe crack length (the crack length is chosen arbitrarily as a = 4 m) which reduces theboundary e�ect to a minimum. Numerically obtained values and theoretical SIFs for di�erentcombinations of the inclined angle � and the applied stresses �11 and �22 are listed in Table7.2. From the table, it can be concluded that the numerical procedure provides an accurateprediction of the mixed mode SIFs arising from a crack located in a biaxially loaded stress�eld.

Table 7.2: SIF in a mixed mode stress �eld.Theoretical Numerical J-EDI

� �11 �22 KI KII KI KII

[0] [MPa] [MPa] [MPapm] [MPa

pm] [MPa

pm] [MPa

pm]

120 0.0 100.0 265.0 -153.0 263.0 -149.060 0.0 100.0 265.0 153.0 262.0 148.0120 100.0 0.0 88.6 153.0 87.7 -148.060 100.0 0.0 88.6 -153.0 87.4 148.0


Figure 7.15: Obtained crack paths forcrack along a butt weld.

Figure 7.16: Crack paths obtained bySumi and Yamamoto [122].

7.6.4 Crack Propagation along a Butt Weld

Brittle fracture often initiates from weld defects associated with conditions such as undercutand lack of penetration. When a crack is initiated along a weld line, it is a�ected byvarious conditions like a) the applied external stress b) the welding residual stresses andc) the degradation of the heat-a�ected zone (HAZ). A signi�cant degradation (reduction offracture toughness) of the material in the HAZ can be found in weldings of high-strengthsteel. A crack initiated along a weld line in a high-strength steel is therefore most likelyto propagate along the weld. The decrease of fracture toughness along the HAZ is lesspronounced in structures made of mild steel, and it has been experimentally observed thata brittle crack initiated along a weld line turns and propagates into the base material [122].

A simpli�cation of the latter case is here used to illustrate the capabilities of the simula-tion procedure. An initial crack with a length of 100 mm is introduced in the zone betweenthe HAZ and the base material in a square plate of 500 mm in length. A longitudinal self-equilibrating welding residual stress is applied perpendicularly to the weld line (see Figure7.15), while the transverse residual stresses arising from the weld are neglected in the presentcase. The isotropic material properties are 0.3 and 206 GPa for Poissons and Youngs modu-lus respectively. Simulations were carried out for three levels of external stresses �a appliedto the two edges perpendicular to the weld. In the simulation, the crack tip is given a smallinitial kink of 2.0 degrees towards the base material. The obtained crack paths are plotted inFigure 7.15, from which it is seen that, in all the cases, the crack turns to the base materialand that the shape of the crack path depends on the applied external stress.

Sumi and Yamamoto [122] carried out simulations for the same con�guration using themethod where the stress state in the vicinity of the crack tip is determined by a superposition


of an analytical and a �nite elements solution. Their simulated results are depicted in Figure7.16. It is seen that the crack paths predicted by the two di�erent methods are in goodagreement.

7.6.5 Mixed Mode Crack Propagation in Residual Stress Fields

Experimental results recently presented by Sumi et al. [125] will be used to evaluate thenumerical model for prediction of mixed mode fatigue crack propagation in residual stress�elds. An outline of the experiments is given here (for details see [125]). Crack growthdata including crack path measurements was obtained for crack propagation in welded platestructures consisting of an I-pro�le with a curve bar welded to the upper ange by a double�llet weld. A sketch of the specimens, which were made of ship structural steel of classKA36, and the rest of the experimental set-up is given in Figure 7.17. From the sketch it is

Figure 7.17: Sketch of test layout for mixed mode crack propagation in a residual stress �eld.

seen that an initial crack is introduced in the bend between the horizontal and the verticalpart of the bar sti�ener. The specimens were exposed to the cyclic load P at the top of thebar sti�ener and simply supported at the ends of the I-pro�le. In [125] crack growth datawas presented for di�erent notch con�gurations and biaxial stress ratios, with control of thelatter feature by the sti�ness of the I-pro�le relative to the rest of the structure.

The residual stresses arising from the attachment of the bar to the I-pro�le were experi-mentally determined by the conventional sectioning technique for a con�guration called B3.The detailed residual stress distribution presented in [125] shows a relatively high level oflongitudinal compressive residual stresses in the range of -200 MPa to -300 MPa with a peakattainment approximately 30 mm from the edge of the specimen. Normal residual stresses


Figure 7.18: Three-dimensional �nite element mesh used for the simulation.

acting in the vertical direction were found to be negligible for the present experimental set-up[125].

Numerical fatigue crack simulations were carried out for the con�guration B3 with thecyclic load P oscillating between 4.9 kN and 88.0 kN (as in the experimental set-up). Theinitial �nite element mesh used in the simulations is depicted in Figure 7.18 with the cracktip location in the high mesh density zone.

As discussed in Section 6.2, it is not yet clear how to determine the e�ective part of the�KII in the presence of crack closure. Two interpretations were applied in the numericalsimulations: 1) �KII taken as the full range ignoring the friction between the two cracksurfaces and 2) an e�ective �KII,e� obtained by Eq. (6.14) where �KII is assumed not tocontribute through the period of crack closure. The two interpretations can be regarded asbounds for the �KII range. The crack is assumed to be two-dimensional in the simulations,however, the real crack geometry is three-dimensional and rather di�erent crack lengthswere observed for some of the experiments carried out by Sumi et al. [125]. However, theexperimentally obtained crack path for the B3 con�guration showed almost the same crackgrowth rates for both sides of the specimen. A reproduction of the measured crack paths onthe front and back side is given by the two bold curves in Figure 7.19 with the labels A and Brespectively. Simulations with �KII taken as the full range were performed for two di�erentsizes of crack increments (�a1 ' 3�a2) to test the mesh sensitivity of the calculations.These are denoted E and F in Figure 7.19. The curve D represents the reduced �KII whilethe last curve C is the numerically obtained crack path when the e�ect of residual stresses isneglected. It is seen from the �gure that the best agreement between the experimental pathsand the simulated paths is observed when �KII is taken as the full range. The simulationwith the reduced �KII is very close to the crack path determined for the con�gurationwithout residual stresses. The increment size is seen to have local in uence on the simulatedcrack path. But the global pattern is the same for the predictions E and F.

7.7. Summary 121

Figure 7.19: Experimentally obtained and simulated crack paths.

Crack growth curves for the same experimental set-up are presented in Figure 7.20 withcurves A and B being reproductions of measured a-N curves from [125]. The crack growthequations given by Eq. (6.13) and Eq. (6.12) were used for the numerical estimations. It isseen that the negligence of residual stresses (curve C) results in very conservative fatiguelife estimations. The measured a-N curves are for crack lengths longer than 35 mm \sur-rounded" by the curves based on the two di�erent determinations of �KII e�ective. Thefull-range interpretation results in conservative estimations while the opposite is observedfor the reduced �KII method. But, again, the best correlation between experiments andsimulations is obtained for the full-range method. Hence, it can be concluded that the full-range interpretation gives the best description of the experimental crack growth data for theB3 con�guration.

7.7 Summary

Relatively simple procedures for qualitative estimation of longitudinal residual stresses aris-ing from welding and ame-cutting have been outlined. The methods are based on simpleformulas for estimation of the inherent strains in the structures. The inherent strain distri-butions can be used as an input for a linear elastic determination of residual stresses.


Figure 7.20: Experimentally obtained and simulated crack paths relative to the number ofload cycles.

The superposition method has been applied to account for residual stresses in the de-termination of stress intensity factors, and the necessary modi�cation of the J-integral hasbeen introduced. Test examples showed a good agreement between theoretical SIFs and theresults obtained by the numerical procedures applied. The two latter examples showed thatthe numerical crack simulation model provides good crack path estimations for fatigue crackgrowth and brittle fracture close to welds. The mesh density was shown to have local in u-ence on the crack paths, but the global behaviour was not signi�cantly a�ected by changesin crack increment size.

Two di�erent determinations of �KII in the presence of compressive residual stresseswere used in the fatigue crack simulations. The best results were obtained for the \full-range" philosophy. However, more simulations and experiments are needed for a generalconclusion. Other examples of mixed mode crack growth in residual stress �elds have notbeen found in the literature.

On the basis of the presented test examples, it is concluded that the numerical procedureoutlined in Chapter 5, combined with the modi�cations for crack closure and residual stresses,provides a reliable estimation of crack propagation under the in uence of residual stresses,and the model will be applied to the cutouts described in Chapter 3.

Chapter 8

Crack Simulation Procedure Applied

to Cutouts

8.1 Introduction

In this chapter the two-dimensional crack simulation procedure will be used in a comparativestudy of cracks propagating from the cutouts in the conventional and the shape-optimisedslot structures described in Chapter 3. Crack propagation starting from the toe of the bracketused in the conventional design will not be included in the present analysis. Cracks, 1 mmin length, are initiated at points of high stress concentration and a possible crack growthis investigated in an attempt to estimate the relative fatigue strength of the two designs.A qualitative estimation of the residual stress e�ect which arises from the manufacture ofthe slot structure is included. The �nite element models of the slot structures and theapproximations made are discussed in the following section.

8.2 Modelling of the Problem

A good geometrical description of the cutouts is important to the investigations. Hence,the cutouts are modelled with a relatively high-density mesh concentrated in the regionsaround the initial cracks, as depicted in Figure 8.1 and Figure 8.2 for con�gurations withinitial cracks in the bends. The longitudinal transfers load from the side shell into the webframe. The actual stress distribution within the longitudinal is of minor importance to crackspropagating in the web frame. Thus, the body of the longitudinal is modelled by membraneelements, which ensures a proper transfer of sea water pressure into the web frame, whilethe side shell and the ange of the longitudinal are modelled by truss elements in order toreduce the number of nodal points.

123

124 Chapter 8. Crack Simulation Procedure Applied to Cutouts

Figure 8.1: Mesh at the cutout for the con-ventional slot structure with the initial cracklocated in the bend.

Figure 8.2: Mesh at the cutout for the shape-optimised slot structure with the initial cracklocated in the bend.

The bracket in the conventional design will cause unsymmetrical bending, which resultsin out-of-plane deformation of the web frame. The Mode III deformation (see Figure 4.1)can not be accounted for by the simulation procedure and a simpli�cation has to be made.Two brackets, symmetrically located with respect to the web frame and with half of thethickness of the original sti�ener, are used to transfer the load from the longitudinal to theupper part of the web frame so that in-plane deformation of the web is ensured. A 3-Dview of the mesh used for the crack tracing in the conventional slot structure is given inFigure 8.3. The truss elements used for the modelling of the side shell and the ange of thelongitudinal are omitted in the visualisation. It should be noted that the use of the trusselements neglects the connections along the weld seam between the side shell and the webframe.

8.2.1 Loads and Boundary Conditions

The crack simulation procedure is restricted to constant amplitude loading. Hence, a rep-resentative constant cyclic loading has to be determined for the simulations. The results ofthe probabilistic fatigue life estimations presented in Chapter 3 can be used as a startingpoint. In the discussion of Figure 3.16, it was noted that the laden condition accounts forapproximately 90 % of the damage, and from the same �gure it could be concluded thatmost of the damage is caused by the external sea water pressure acting on the side shell.From these observations, it was decided to base the simulations on the vessel being in ladencondition with an appropriate cyclic wave pressure acting on the side shell. The establishedprobabilistic model was used for the determination of an equivalent cyclic wave. The sim-ulation was based on two observations: 1) A non-linear relation exists between the stressrange at the crack tip and the damage to be expected (see e.g. Eq. (3.6) or Eq. (4.23)), and2) the cyclic stress state at the crack tip is strongly related to the sea water pressure on the

8.2. Modelling of the Problem 125

Figure 8.3: Finite element mesh used for the crack simulation in the conventional design.

side shell. These observations lead a determination of an equivalent wave height heq by thefollowing relation:

heq =

"PNi=1(hi)

m

N

#1=m(8.1)

where hi is a wave height simulated by the probabilistic model and N is the number ofsimulation points. The damage caused by large wave heights relative to the damage inducedby small wave heights is controlled by the parameter m. On the basis of the crack growthequations outlined in the Sections 3.2.6 and Section 6.2.1 and the assumptions described inthis section, it was decided to used m equal to 3.0. The outcome of the simulation was awave amplitude of 0.93 m.

An examination of the loads presented in Figure 3.8 reveals that only the three latterload cases are of interest to the simulation of crack propagation in the web frame. Thecontribution from each of these cases is determined on the basis of the equivalent wave. Theinternal pressure from the cargo shows some variation because of the vertical accelerationof the ship. But these variations are relatively small (especially for the midship tank) andthe cargo pressure is assumed to be constant during the simulations. Thus, the resulting


Figure 8.4: Boundary conditions and equivalent wave used in the simulations.

load case consists of a static contribution from internal cargo pressure and uctuating loadscontrolled by the external wave height.

The equivalent wave and the boundary conditions for the conventional design are depictedin Figure 8.4. The web frame is supported at the edge pointing in the direction of the cargotank, and symmetry conditions are applied at the end of the longitudinal (located in themidpoint between two web frames). Similar boundary conditions are applied to the shape-optimised design, except for the support at the end of the brackets.

8.3 Crack Propagation without Residual Stresses

From the von Mises stress plots presented in Chapter 3, it was found that, in the conventionaldesign, cracks are most likely to initiate at the lower part of the bends and at the weld seambetween the longitudinal and the web frame. 1 mm cracks were initiated at these locationsand the simulation procedure was used. But, with the applied cyclic loads, given by theequivalent wave, crack arrest was predicted for both con�gurations at the initial stage. Fora matter of comparison, simulations were carried out with the threshold value equal to zero.Figure 8.5 shows the obtained crack path for the step where the crack initiated in the bendhas propagated 0.047 m, while the obtained crack path for a crack initiated at the weld seamis depicted in Figure 8.6.

The crack initiated in the bend turns and propagates in a direction which is almost per-pendicular to the side shell, while the crack starting at the location of the weld propagatestowards the intersection between the longitudinal and the web frame. The simulation proce-dure allows for tracing of cracks propagating in three-dimensional plate structures. However,

8.3. Crack Propagation without Residual Stresses 127

Figure 8.5: Simulated crack path for the con-ventional structure with a crack in the bend.

Figure 8.6: Simulated crack path for a crackinitiated at the weld seam.

the cracks are assumed to propagate in a two-dimensional plane which in the present caseis de�ned by the undeformed web frame. A possible crack propagation from the web intothe longitudinal is therefore not within the scope of the present formulation. Thus, the sim-ulation is terminated when the predicted crack path enters the domain of the longitudinal.Another limitation is that the material of the structure is assumed to be isotropic. Hence,material changes due to manufacturing procedures are not taken into account.

Figure 8.7: Simulated crack path with re-duced mode II range.

Figure 8.8: Simulated crack path with fullmode II range.

In the shape-optimised structure a crack is initiated in the bend of the cutout where themain stress concentration is found. Crack arrest was also predicted at the initial stage forthis con�guration and again a simulation was carried out with the threshold value set tozero. Partial closure was present during the simulation which reduced the e�ective mode Irange. The crack path presented in Figure 8.7 is for a reduced �KII range while the resultof a simulation with full �KII range is depicted in Figure 8.8. The crack paths are verysimilar and both simulations predict full crack closure for the depicted locations of the cracktips (KI is not above zero during the whole load cycle). But the crack trace with full �KII


range stops one iteration after the trace with the reduced mode II range (for the same crackincrement sizes).

Figure 8.9: Equivalent stress intensity range �Keq as a function of the crack length a.

The equivalent mixed mode stress intensity range (see Eq. (4.28)) is plotted as a functionof the increasing crack length for the four simulated crack paths in Figure 8.9. The equivalentstress intensity range is below the threshold values given in Section 6.2.1 for the whole rangeof simulated paths. Nevertheless, when the possible crack arrest is disregarded, an interestingtrend can be observed. At the initial stage of the crack propagation the stress intensity rangeis relatively large for the optimised structure, however, it drops to zero after a short distanceof crack propagation and the crack faces are closed. The equivalent stress intensity rangefor the conventional design starts at a lower level, but it increases continuously during thecrack propagation (see Figure 8.9). Thus, closure of crack surfaces is not observed for thiscon�guration.

8.4 Including the E�ect of Residual Stresses

This section presents a qualitative estimation of how residual stresses can a�ect the crackpropagation in the two types of structures. Residual stresses inherent from the manufactureof the structures are considered while other sources of residual stresses, such as initial stressesdue to the rolling of the plates, are disregarded.

8.4.1 Modelling of Residual Stresses

The longitudinal to web frame connections are complicated welded structures as it can beseen from Figure 8.10 where the bold lines indicate either a weld seam or a thermally cut edge.

8.4. Including the E�ect of Residual Stresses 129

The estimation of the residual stresses due to the welding of the slot structures is based onthe inherent strain procedure outlined in Section 7.2.2. The welding parameters needed forthe calculation are listed in Table 8.1 with the labels corresponding to the numbers in Figure8.10. Residual stresses originating from the manufacture of the cutout are estimated by the

Figure 8.10: Identi�cation of welding and cutting lines in the slot structure (conventionaldesign).

experimentally obtained Eq. (7.9), which is valid for ame-cut edges. Plasma-cut edges,with a di�erent residual stress distribution because of the heat input, are not considered.

Table 8.1: Welding parameters

Welding speed [cm/min] 42.0

Voltage [V] 27-28Current [A] 270-280Welding speed [cm/min] 17.0Voltage [V] 25Current [A] 215-220Welding speed [cm/min] 110Voltage [V] 27� 3Current [A] 750 � 50

Figure 8.11 shows the distribution of one of the resulting principal strain componentswhich is used as input for the calculation of the residual stresses in the conventional slotstructure. The inherent strains from the di�erent contributions have simply been added.


Figure 8.11: Inherent strains used as input for the determination of residual stresses in theconventional structure (multiplyed by 103).

This causes some errors where two distributions meet (strains from two di�erent sourcescannot simply be added [47], however, a theoretical estimation is rather di�cult). Anothersource of inaccuracy is that the applied inherent strain procedure is not strictly valid atthe ends of the welds where the boundary e�ects manifest themselves. For the presentcomparative study, these inaccuracies have been accepted. Nevertheless, a more accuratedetermination of the residual stress distributions could be obtained by special-purpose pro-grams such as SYSTUS+ [126] and the obtained stress distributions could be applied asinput for the crack simulation procedure.

The web frames considered are made of mild steel with a yield stress level close to 235MPa while the yield strength of the weld material is assumed to be 50 % higher (as suggestedin [146]). In the web frame, the largest inherent strains (negative) can therefore be found atthe location of the weld lines which have the labels 1 and 3 in Figure 8.10.

The deformation of the cutout imposed by the inherent strains can be obtained by linearelastic �nite element calculations. The ame-cutting causes the edge to contract and tensilestresses at the yield stress level can be found along the edge. Similar contractions occur bythe welding of the longitudinal and the bracket. Figure 8.12 shows the deformation patternfor the conventional structure when it is exposed to the inherent strain distribution depicted


in Figure 8.11. The deformations are relatively symmetric with respect to the longitudinaldespite the slightly non-symmetric input of inherent strain caused by the di�erence in meshdensity on the two sides of the longitudinal.

Figure 8.12: Deformation of the conventional structure when it is exposed to the inherentstrain distribution in Figure 8.11. The deformations are multiplyed by a factor of 200.0.

Figure 8.13 shows the normal residual stress component in the direction perpendicularto the weld. The contraction of the weld seams transforms the tensile stresses from the ame-cutting into compressive normal stresses at the connection between the web frame andthe longitudinal.

Tensile stresses with the main components parallel to the edge are present in the bends.The ame-cutting results in residual stresses at the yield level of the cut edges. When thecontraction of the welds is superposed, the residual stresses predicted by the linear elasticanalysis exceed the yield stress of the material at the bends. This zone is fortunately limitedto a narrow band in the middle of the bend. In the present formulation, the stresses arecut o� at the yield stress level by use of the von Mises stress criterion. For a more accuratedetermination, the redistribution of stresses due to plasticity should be included.

The discussion of residual stresses has focused on the conventional slot structure. Anal-ogous calculations were performed for the shape-optimised structure. The obtained residualstress distribution is presented in the form of von Mises stresses in Figure 8.15. The elimina-tion of the bracket changes the residual stress distribution at the upper part of the cutout,otherwise no signi�cant changes can be found between the residual stress distribution foundin the conventional and the shape-optimised slot structures.


Figure 8.13: The component of the normal residual stresses in the direction perpendicularto the welds.

Figure 8.14: The resulting von Mises stress distribution for the conventional cutout.


Figure 8.15: The resulting von Mises stress distribution for the shape-optimised cutout.

8.4.2 Result of Crack Path Simulation

Cracks were initiated at the same location as for the simulations without residual stresses. Aninteresting observation was made for the crack initiated close to the weld seam. Compressivenormal residual stresses forced the crack surfaces together throughout the whole load cycleand caused crack arrest at the initial stage.

The residual stresses due to welding and thermal cutting were assumed to be static andonly a�ected by the propagating crack. Thus, the cyclic stress range was not changed bythe introduction of residual stresses, and the application of the cyclic loads given by theequivalent wave still resulted in crack arrest at the initial stage for the con�gurations withinitial cracks in the bends. Again, the threshold value was set to zero and \crack stop" wasde�ned as the mode I stress intensity being zero during a whole load cycle. Crack simulationswere carried out and the patterns found in the simulations without residual stresses could berecognised. For the conventional slot structure almost no in uence was observed from thechange in R-value. Figure 8.16 shows the obtained path where the crack has been allowedto grow to a length of 0.15 m. The slow increase in stress intensity range observed in thesimulations without residual stresses (see Figure 8.9) was also found here and \crack stop"was not observed. The optimised cutout exhibited some in uence of the changed R-level,which resulted in an increased crack length compared to the simulations without residualstresses. But again closure of the faces was detected, in the present case for the crackcon�guration depicted in Figure 8.17.


Figure 8.16: Simulated crack path for opti-mised cutout with residual stresses.

Figure 8.17: Simulated crack path for con-ventional cutout with residual stresses.

8.5 Comments and Summary

In an evaluation of the in uence of residual stresses, the assumptions made in the qualitativeprediction of the stress �elds should be kept in mind. Inaccuracies are to be expected at theends of the welds and where the e�ects of two manufacturing procedures interfere.

From the simulations carried out for the conventional structure, it was observed thatcompressive stresses at the location of the weld reduce the risk of crack propagation in thisarea, which is an important observation because imperfections from the welding are oftenfound in this region.

The application of an \equivalent wave concept" resulted in prediction of crack arrest atthe initial stage of the simulations. Figure 8.18 shows the distribution of the simulated waveheights with the equivalent wave height indicated by the dashed line. From the �gure it canbe seen that waves which are signi�cantly larger than the equivalent wave can be expectedduring the lifetime of the vessel. The actual wave heights which result in propagation ofcracks initiated in the two structures have not been determined. But, the stress intensityranges depicted in Figure 8.9 together with the threshold values outlined in Section 6.2.1indicate that crack propagation may occur during the service of the vessel. The e�ect of peakloads can not be captured by the established high-cycle fatigue model. A more sophisticatedmodel which includes sequence e�ects would have to be applied for an investigation of crackpropagation caused by peak loads.

The prediction of crack arrest at the initial stages makes it di�cult to predict the rela-tive fatigue strength of the two structures. Nevertheless, crack simulations performed withthreshold values equal to zero indicate that the initial stress intensity range for the shape-optimised cutout is larger than for the conventional cutout (due to the larger load carriedby the weld seam between the longitudinal and the web frame) but that it drops to zeroat a given distance. The stress intensity range for the conventional structure starts at a

8.5. Comments and Summary 135

Figure 8.18: Distribution of simulated wave heights.

lower range, but it increases continuously as the crack propagates. The same trend is to beexpected at a stress level above the threshold limit. Thus, the optimised slot design seems tobe superior to the conventional design. Finally, crack propagation initiated at the toe of thebracket used in the conventional design should enter a complete evaluation of the relativefatigue strength of the two designs.

Chapter 9

Conclusion and Recommendations for

Further Work

9.1 Conclusion

The large number of longitudinal to web frame connections present an essential part of theproduction costs of a very large crude oil carrier (VLCC). The slot structures are complicatedwelded details which are exposed to dynamic loads during the service of the vessels and ifnot adequately designed, signi�cant fatigue cracking may take place and cause major repaircosts as in the case of the early cracking of the second-generation VLCCs. The objective ofthe present work has been to develop and investigate a new type of slot design suitable forapplication of welding robots. This included establishing a probabilistic model for non-linearestimation of fatigue lives and a general-purpose mixed mode crack simulation procedure forprediction of curved crack propagation in welded structures. The models have been appliedin a comparison of the fatigue strength of the new slot structure relative to the strength ofa conventional design.

In conventional slot structures, cracks can often be found at the toe and the heel of thebracket used to transfer load from the pressure on the side shell into the web frame. In thenew design, the sti�ener is removed in order to avoid these cracks and to ease the applicationof welding robots (no manual �tting of the bracket). The load to be carried by the weldseam between the web frame and the longitudinal is thus increased and the cutout becomes ahard-spot area. A �nite element based shape optimisation was applied to reduce the stressesat the cutout. Before the shape optimisation, the main stress concentrations were at thewelds which are known to be crack initiators. The shape optimisation reduced the stresslevel by a factor of 1.6 in this region and in the new design the main stress concentrationswere found in the bends away from the welds.

Before the introduction of signi�cant structural changes at locations known to be proneto fatigue cracking, it is good design practice to estimate the fatigue life of the new design.

137

138 Chapter 9. Conclusion and Recommendations for Further Work

A fatigue damage model with a probabilistic load generation based on the sailing route ofthe vessel was established for that purpose. By applying the quasi-stationary narrow-bandmodel, it was possible to include the non-linear e�ect of intermittent submergence of the sideshell found in the zone close to the load line. Fatigue lives of the new and the conventionaldesigns were predicted by the probabilistic model for details located approximately 2 metresbelow the still water load line in the region where the longitudinal to web frame connectionsare most prone to fatigue. The estimated fatigue lives of the weakest points of the shape-optimised cutout were found to be approximately of the same order of magnitude, close to30 years. For the conventional structure, it was predicted that cracks are most likely toinitiate at the toe of the bracket with an estimated lifetime of 5 years, i.e. 6 times less thanthe predicted lifetime of the new design.

Cracks can be initiated at imperfections caused by welding and thermal cutting of edges.Conventional fatigue damage approaches are normally based on the assumption that if sub-sequent crack propagation occurs, it is in a self-similar manner. However, branched crackpropagation can be found in structures with residual stresses or in details exposed to bi-axial loading where curved crack propagation may have a signi�cant in uence on the crackgrowth rate. A two-dimensional multipurpose crack propagation procedure has been estab-lished in order to investigate mixed mode crack propagation in welded structures subjectedto high-cycle loads. The method was based on a step-by-step �nite element procedure withcontinuous remeshing of the domain surrounding the propagating crack. A two-dimensionalfree mesh generator was established for the meshing of the crack domain. The mesh genera-tor, based on the advancing-front method, was found to be stable and robust and thus wellsuited for support of discrete crack propagation. Special emphasis was put on the numericalevaluation of the mixed mode stress intensity factors, which are the controlling parametersin a linear elastic fracture mechanical approach. The formulated model allows for determi-nation by the displacement correlation technique as well as the J-integral in the equivalentdomain integral formulation with decomposition applied. Three common methods for esti-mation of crack propagation angles were employed. The simulation procedure was validatedby comparison with experimental and numerical data found in the literature covering boththe in uence of holes and biaxial loading. A good correlation was obtained. The simulationsalso revealed that the choice of procedures for determination of crack propagation angle andstress intensity factors does not in uence signi�cantly the prediction of the crack paths. Itis therefore concluded that the presented crack simulation scheme is able to handle crackpropagation in general two-dimensional domains.

Crack closure is when the two crack surfaces come into contact. The phenomenon a�ectsthe crack growth and it becomes of major importance when cracks are located in compressiveresidual stress �elds. The occurrence of closure is in uenced by a large number of factors suchas plasticity, oxide on the crack surface and roughness of the crack faces. The complexityof closure mechanisms does not allow for direct modelling of the phenomenon by the �niteelement method. The e�ective stress intensity range is de�ned as the part of the load cyclewhich causes damage to the material. Crack closure was modelled by empirical formulaswhich related the e�ective stress intensity range to the stress state at the crack tip. Anevaluation of methods for determination of e�ective mixed mode stress intensity ranges left

9.1. Conclusion 139

some doubt concerning the determination of the e�ective mode II range. Two models wereused: 1) An e�ective range assumed to follow the e�ective mode I range, and 2) the wholemode II range regarded as e�ective.

Two simple procedures for qualitative estimations of residual stresses caused by ame-cutting and welding have been outlined with the latter approach based on the inherent strainmethod. Obtained strain distributions can be used as an input for a linear elastic determi-nation of residual stresses in the structures, or a known residual stress distribution can begiven as an input. To account for the residual stresses in the determination of the stressintensity factors, the superposition method was employed. The capabilities of the proce-dure were tested by several benchmark examples which revealed that good estimations wereprovided for high-cycle fatigue crack growth as well as brittle fracture close to welds. Twodi�erent determinations of the mode II range in the presence of compressive residual stresseswere tested against experimental results for cracks propagating in a three-dimensional platestructure. The best result was obtained for the \full-range" philosophy. However, moresimulations and experiments are needed for a general conclusion on the e�ective mode IIrange.

The crack simulation procedure was applied to the conventional and the shape-optimisedslot structures. Residual stresses were estimated on the basis of relatively simple inherentstrain distributions. From the simulations, it was observed that compressive residual stressesat the location of the weld reduce the risk of crack propagation in this area, which is animportant observation because imperfections from welds are often found in this region. Theassumptions made in the qualitative prediction of the inherent strain �elds should, however,be kept in mind in evaluations of the in uence of residual stresses.

An equivalent wave height was determined by the probabilistic model described in Chap-ter 3. However, the stress range obtained by applying the equivalent wave resulted in pre-diction of crack arrest at the initial stage, which makes it di�cult to conclude that one ofthe designs is superior to the other with respect to fatigue. Nevertheless, the crack simula-tions performed with threshold values equal to zero indicate that the initial stress intensityrange for the optimised cutout is larger than for the conventional cutout, but it drops tozero at a given distance. The stress intensity range for the conventional structure starts at alower range, but it increases continuously as the crack propagates. The same trend is to beexpected at a stress level above the threshold limit. Thus, the optimised slot design seemsto be superior to the conventional design.

140 Chapter 9. Conclusion and Recommendations for Further Work

9.2 Recommendations for Further Work

The outlined probabilistic model and the crack simulation procedure are believed to covermain issues of fatigue assessment. However, improvements of the models could be made andadditional e�ects could be included. In the work on the probabilistic model further researchmight deal with

� taking into account the e�ect of corrosion, which is believed to be the single mostimportant cause of failure of ship structures,

� investigation of di�erent longitudinal pro�le types such as L- and bulb pro�les, or othervertical positions of the T-pro�le,

� taking into account e�ects of the load sequence and the mean stress level,

� use of three-dimensional �nite element models and strip programs for the determinationof loads.

Further work on the simulation procedure can be divided into subjects which involve minore�orts or changes of the established program

� control of the crack increment size based on the changes in stress intensity at the cracktip,

� multicrack propagation,

� a comparison of residual stresses estimated by the inherent strain procedure with distri-butions predicted by state-of-the-art programs which include plasticity and thermallycaused material changes,

� investigation of crack propagation initiated at the toe of the bracket,

and issues which require a major e�ort, such as

� variable amplitude loading (related to sequence-e�ects),

� establishing a model for determination of residual stresses from plasma cutting,

� better understanding of crack closure in mixed mode stress �elds,

� taking into account material changes due to welding and thermal cutting,

� including the residual stresses from the rolling of the plates,

� redistribution of residual stresses during loading of the structures.

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Appendix A

Singular Triangular Quarter-Point

Elements

This appendix outlines shortly how the singular stress state is obtained along the crack faceof a singular triangular quarter-point element. For a general proof of singularity within theelement see Barsoum [15].

A normal isoparametric 8-noded quadratic element is used for the formulation. Theelement sti�ness is well documented for the isoparametric elements and in accordance withthe notation of Zienkiewicz [154] an 8-noded plane isoparametric element is mapped into thenormalised space (�; �); (�1 � � � 1;�1 � � � 1) through the transformation

x1 =8Xi=1

Ni(�; �)x1;i

x2 =8Xi=1

Ni(�; �)x2;i

Ni = [(1 + ��i)(1 + ��i)� (1� �2)(1 + ��i)� (1� �2)(1 + ��i)]�2i �

2i =4+

(1� �2)(1 + ��i)(1� �2i )�2i =2 + (1� �2)(1 + ��i)(1� �2i )�

2i =2

(A.1)

where Ni is the shape function corresponding to the node i, whose coordinates are (x1;i; x2;i)in the global x1-x2 system and (�i; �i) in the normalised system. Figure A.1 shows thecollapsed element, in which

x1;1 = x1;4 = x1;8 = 0; x1;5 = x1;7 =h

4; x1;2 = x1;3 = x1;6 = h

and

x2;1 = x2;4 = x2;6 = x2;8 = 0; x2;5 = �x2;7 = � l

4; x2;2 = �x2;3 = �l

(A.2)

153

154 Appendix A. Singular Triangular Quarter-Point Elements

Figure A.1: Collapsed quarter-point element and parent element.

By substitution Eq. (A.2) into Eq. (A.1) and collection of the terms, the following expressionsare derived:

x1 =h

4(1 + �)2

and (A.3)

x2 =l

4�(1 + �)2

For simplicity, only the displacement of the edge with the points 1,2, and 5 is evaluated.Here the displacement u1 are given by

u1 = �12�(1� �)u1;1 +

1

2�(1 + �)u1;2 + (1� �2)u1;5 (A.4)

If the displacement is written in terms of x1:

u1 = �12

��1 + 2

rx1h

� �2� 2

rx1h

�u1;1 +

1

2

��1 + 2

rx1h

� �2

rx1h

�u1;2 +�

4x1h� 4

x1h

�u1;5 (A.5)

the strain in the x1-direction can be found by use of the chain rule,

"11 =@u1@x1

=@�

@x1

@u1@�

= �12

"3px1h

� 4

h

#u1;1 +�1

2

"� 1p

x1h+4

h

#u1;2 +

"2px1h

� 4

h

#u1;5 (A.6)

It is seen that strain singularity along the side 1-2 is of the type 1=pr, which is equal to the

theoretical strain �eld for linear elastic materials.

Appendix B

Numerical Examples

Stress intensity factors calculated by the procedures described in Section 5.3 are in thissection compared to di�erent benchmark examples. For the simple mode I case, stressintensity factors collected in compendiums are used for the evaluation. The collected stressintensity factors are typically of the form:

K� = �0p�af(a; w; h) � = I or II (B.1)

where �0 is the far �eld stress, a is the crack length. The width and the height of thespecimen are denoted by w and h respectively, while the e�ect of the boundaries is takeninto account by the parameter f which is equal to one for a crack in an in�nite plate.

Stress intensity factors obtained by the numerical procedure are in the mixed mode casecompared to a theoretical result, data obtained by Aliabadi et al. [1] using boundary elementanalysis, and an experiment where the path of a propagating crack has been measured.

B.1 Single- and Double-Edge Cracks, Mode I

The �rst state considered is a single-edge crack in a plate subjected to uniform tensile stressesacting perpendicularly to the direction of the crack. The second case is similar to the �rststate just with a double-edge crack con�guration. The two states are illustrated in FigureB.1 and Figure B.2 respectively.

155

156 Appendix B. Numerical Examples

Figure B.1: Single-edge crack, Mode I. Figure B.2: Double-edge crack, Mode I.

Plane stress conditions are assumed for the two cases and the following data is used forthe calculations (arbitrarily chosen):

� Field stress �0 = 960 Pa.

� Height = 2h = 20 m.

� Width = 2w = 20 m.

� No body forces.

The stress intensity factors determined for di�erent crack sizes are presented in Figure B.3and Figure B.4. A good correlation is obtained between the stress intensity factors calculatedby the three methods presented in Section 5.3 and the stress intensity factors based onEq. (B.1). The best results are produced for the single-edge specimen, here all the numericalresults are within a 3 % error in a comparison to Eq. (B.1). The numerical results fromthe specimen with the double-edge crack con�guration show a �ne agreement for relativelysmall crack lengths. But, a small divergence starts as the crack length increases, and themaximum error is about 4 % for the J-contour integral method when a/w = 0.4.

Figure B.3: Single-edge crack, Mode I. Figure B.4: Double-edge crack, Mode II.

B.2. Single-Edge Crack in Mixed Mode Condition 157

Figure B.5: Con�g-uration for inclinedsingle-edge crack.

Figure B.6: Obtained results for inclined single-edge crack con�g-uration, � = 67:5o.

B.2 Single-Edge Crack in Mixed Mode Condition

By application of a boundary element method together with Bueckner weight functions,Aliabadi et al [1] obtained very accurate results for the mixed mode example describedin Section 7.6.3. Thus, their results for a specimen with the single-edge inclined crackcon�guration illustrated in Figure B.6 are here used as a benchmark example. In the presentcase, the comparison is carried out for the following con�guration (arbitrarily chosen):

� Crack propagation angle � = 67:50.

� Field stress �0 = 960 Pa.

� Width = w = 10 m.

The outcome of the calculations is depicted in Figure B.6. From the �gure it is observedthat all the presented stress intensity factors are almost identical with the results of Aliabadiet al [1] for both mode I and mode II.


B.3 Comparison with Experimental Results

With the theory outlined in Chapter 4, it should be possible to carry out a crack pathsimulations using one of the described methods to calculate the stress intensity factors andthe relation given by Eq. (4.11) to determine the crack propagation direction.

Figure B.7: Test specimen.

The starting point for simulations is a plate which is to be welded to another plate by adouble �llet weld. Full penetration is not obtained during the welding and a crack is formedas illustrated in Figure B.7. In this primitive simulation some assumptions are made: 1)there are no residual stresses in the structure, 2) the weld and the base material can beconsidered as a homogeneous body, and 3) the structure is only exposed to tensile forces.The last assumption makes it possible to disregard the e�ect of crack closure. The simulationwas carried out for a structure,

� made of high-tensile steel st 52,

� subjected to a sinusoidal tensile force F , ensuring that crack closure would not occur.

Lyngbye and Nickelsen [75] performed some experiments for the con�guration describedabove. A crack path was simulated using the displacement correlation technique for thedetermination of stress intensity factors and the maximum tangential stress criterion forthe determination of the crack propagation angle. A comparison between the simulatedand experimentally obtained paths is given in Figure B.8. It is seen that the simulation\underestimates" the propagation direction, but the di�erence is relatively small.

B.3. Comparison with Experimental Results 159

Figure B.8: Experimental and simulated crack path.

Ph.d. ThesesDepartment of Naval Architecture and O�shore Engineering, DTU

1961 Str�m-Tejsen, J.: \Damage Stability Calculations on the ComputerDASK".

1963 Silovic, V.: \A Five Hole Spherical Pilot Tube for three DimensionalWake Measurements".

1964 Chomchuenchit, V.: \Determination of the Weight Distribution ofShip Models".

1965 Chislett, M.S.: \A Planar Motion Mechanism".

1965 Nicordhanon, P.: \A Phase Changer in the HyA Planar Motion Mecha-nism and Calculation of Phase Angle".

1966 Jensen, B.: \Anvendelse af statistiske metoder til kontrol af forskelligeeksisterende tiln�rmelsesformler og udarbejdelse af nye til bestemmelseaf skibes tonnage og stabilitet".

1968 Aage, C.: \Eksperimentel og beregningsm�ssig bestemmelse af vind-kr�fter p�a skibe".

1972 Prytz, K.: \Datamatorienterede studier af planende b�ades fremdriv-ningsforhold".

1977 Hee, J.M.: \Store sideportes ind ydelse p�a langskibs styrke".

1977 Madsen, N.F.: \Vibrations in Ships".

1978 Andersen, P.: \B�lgeinducerede bev�gelser og belastninger for skib p�al�gt vand".

1978 R�omeling, J.U.: \Buling af afstivede pladepaneler".

1978 S�rensen, H.H.: \Sammenkobling af rotations-symmetriske og generelletre-dimensionale konstruktioner i elementmetode-beregninger".

1980 Fabian, O.: \Elastic-Plastic Collapse of Long Tubes under CombinedBending and Pressure Load".

1980 Petersen, M.J.: \Ship Collisions".

1981 Gong, J.: \A Rational Approach to Automatic Design of Ship Sec-tions".

1982 Nielsen, K.: \B�lgeenergimaskiner".

1984 Rish�j Nielsen, N.J.: \Structural Optimization of Ship Structures".

1984 Liebst, J.: \Torsion of Container Ships".

1985 Gjers�e-Fog, N.: \Mathematical De�nition of Ship Hull Surfaces usingB-splines".

1985 Jensen, P.S.: \Station�re skibsb�lger".

1986 Nedergaard, H.: \Collapse of O�shore Platforms".

1986 Junqui, Y.: \3-D Analysis of Pipelines during Laying".

1987 Holt-Madsen, A.: \A Quadratic Theory for the Fatigue Life Estima-tion of O�shore Structures".

1989 Vogt Andersen, S.: \Numerical Treatment of the Design-AnalysisProblem of Ship Propellers using Vortex Latttice Methods".

1989 Rasmussen, J.: \Structural Design of Sandwich Structures".

1990 Baatrup, J.: \Structural Analysis of Marine Structures".

1990 Wedel-Heinen, J.: \Vibration Analysis of Imperfect Elements in Ma-rine Structures".

1991 Almlund, J.: \Life Cycle Model for O�shore Installations for Use inProspect Evaluation".

1991 Back-Pedersen, A.: \Analysis of Slender Marine Structures".

1992 Bendiksen, E.: \Hull Girder Collapse".

1992 Buus Petersen, J.: \Non-Linear Strip Theories for Ship Response inWaves".

1992 Schalck, S.: \Ship Design Using B-spline Patches".

1993 Kierkegaard, H.: \Ship Collisions with Icebergs".

1994 Pedersen, B.: \A Free-Surface Analysis of a Two-Dimensional MovingSurface-Piercing Body".

1994 Friis Hansen, P.: \Reliability Analysis of a Midship Section".

1994 Michelsen, J.: \A Free-Form Geometric Modelling Approach with ShipDesign Applications".

1995 Melchior Hansen, A.: \Reliability Methods for the Longitu-dinalStrength of Ships".

1995 Branner, K.: \Capacity and Lifetime of Foam Core Sandwich Struc-tures".

1995 Schack, C.: \Skrogudvikling af hurtigg�aende f�rger med henblik p�as�dygtighed og lav modstand".

1997 Cerup Simonsen, B.: \Mechanics of Ship Grounding".

1997 Riber, H.J.: \Response Analysis of Dynamically Loaded CompositePanels".

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Fatigue Crack Initiation and Growth in Ship Structures

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